Download Magnetic, chemical and rotational properties Colin Peter Folsom

Magnetic, chemical and rotational properties
of the Herbig Ae/Be binary system HD 72106
by
Colin Peter Folsom
A thesis submitted to the
Department of Physics, Engineering Physics and Astronomy
in conformity with the requirements for
the degree of Master of Science
Queen’s University
Kingston, Ontario, Canada
September 2007
c Colin Peter Folsom, 2007
Copyright Abstract
Recently, magnetic fields have been detected in a handful of pre-main sequence Herbig
Ae and Be (HAeBe) stars. This hints at an evolutionary connection between magnetic
HAeBe stars and Ap and Bp stars. Ap/Bp stars are magnetic chemically peculiar
main sequence stars, whose origins remain poorly understood.
In this context the HD 72106 system, a very young double star, is particularly interesting. HD 72106 consists of an intermediate mass primary with a strong magnetic
field and an apparently non-magnetic Herbig Ae secondary. In this work we examine
20 high-resolution spectropolarimetric observations of HD 72106A and B, obtained
with the ESPaDOnS instrument at the Canada-France-Hawaii Telescope. We find
that the HD 72106 system is a true binary, based on the components’ positions in
space, locations on the H-R diagram, and dynamical properties. For the primary we
determine an effective temperature Teff = 11000 ± 1000 K and a mass of 2.4 ± 0.4
M ; while for the secondary we find Teff = 8750 ± 500 K and M = 1.9 ± 0.2 M .
Through detailed spectral modeling, strong chemical peculiarities characteristic
of Ap/Bp stars are found in the primary. Abundances of 10 times the solar value are
found for Fe, Si and Ti, with Cr 100 times solar and Nd ∼ 1000 times solar. Helium is
found to display less than 1/10 the solar abundance. By contrast, detailed spectrum
modeling of the secondary shows that it possesses approximately solar abundances.
i
The rotation period and magnetic field geometry of the primary are investigated in
detail. A remarkably short rotation period of 0.63995 ± 0.00014 days is derived. A
dipole magnetic field geometry is found, with Bp = 1300 ± 100 G, β = 60 ± 5◦ ,
and i = 23 ± 11◦ . Doppler Imaging of the surface distribution of Si, Ti, Cr, and Fe
is performed for the primary, and strong inhomogeneities are found. All four maps
present similar abundance patterns, with a large spot near the positive magnetic pole.
Implications of these results, particularly with respect to the stage of evolution at
which chemical peculiarities arise, are discussed.
ii
Co-Authorship
The Doppler Imaging results for HD 72106A presented in this thesis were derived in
collaboration with Dr. O. Kochukhov. This technique is discussed in Section 3.7, and
the results are presented in Section 4.4. The spectral lines used for Doppler Imaging
were identified and prepared for use by the author of this thesis, as were the Least
Squares Deconvolution profiles used in the later part of the analysis. The fundamental properties of the star and the initial model parameters were also determined
by the author. The execution of the Doppler Imaging code was performed by Dr.
Kochukhov. Discussions with Dr. Kochukhov were critical for the interpretation of
the Doppler Imaging presented here. Additionally, Dr. Kochukhov provided useful
input regarding the rotation period of HD 72106A, with respect to the physicality of
several different possible phasings of LSD profiles.
iii
Acknowledgments
Firstly I would like to thank my supervisors Gregg Wade and Dave Hanes, whose
knowledge, assistance and encouragement have been absolutely essential for this thesis. I would also like to thank them for their relaxed attitude towards my ‘mornings’,
which has been essential for my continued mental health.
I am greatly indebted to James Silvester, Jenny Power, and Jason Grunhut. Their
assistance, support and comradery has been indispensable, and this thesis could not
have been completed without them.
A huge thank you goes to my family for their unflagging support and encouragement,
as well as for the occasional free lunch.
I am very grateful to Oleg Kochukhov for his aid with the Doppler imaging of HD
72106A, and for fruitful discussions regarding the rotation period of that star.
Thanks are in order for observations of HD 72106A and B obtained by Evelyne
Alecian, Claude Catala, Jason Grunhut, James Silvester, and Gregg Wade.
Finally I would like to thank the Queen’s Astro grads, Jennifer Shore, Shannon Nudds,
Kathryn Bale, and Dominic Drouin for helping to keep the sanity.
iv
Table of Contents
Abstract
i
Co-Authorship
iii
Acknowledgments
iv
Table of Contents
v
List of Tables
viii
List of Figures
ix
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
The Diversity of Stars . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3
Magnetism in Intermediate Mass Stars . . . . . . . . . . . . . . . . .
17
1.4
Herbig Ae and Be stars . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.5
HD 72106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Chapter 2: Observations . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.1
Observing Stellar Magnetic Fields . . . . . . . . . . . . . . . . . . . .
33
2.2
Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
v
2.3
Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.4
Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Chapter 3: Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.1
Basic Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . .
67
3.2
Binarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.3
Spectrum Synthesis and Fitting . . . . . . . . . . . . . . . . . . . . .
81
3.4
Least Squares Deconvolution . . . . . . . . . . . . . . . . . . . . . . .
89
3.5
Magnetic Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
94
3.6
Period Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.7
Doppler Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Chapter 4: Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.1
Surface Chemical Abundances . . . . . . . . . . . . . . . . . . . . . . 113
4.2
Rotation Period of the Primary . . . . . . . . . . . . . . . . . . . . . 127
4.3
Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.4
Surface Abundance Geometry . . . . . . . . . . . . . . . . . . . . . . 138
Chapter 5: Summary, Discussion and Conclusions
. . . . . . . . . . 147
5.1
Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2
Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 150
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Appendix A: CFHT Observing Proposal
. . . . . . . . . . . . . . . . 171
Appendix B: Computer Programs Written . . . . . . . . . . . . . . . 179
vi
B.1 Continuum Normalization: norm.f . . . . . . . . . . . . . . . . . . . . 179
B.2 Period Searching with Stokes I and V LSD
Profiles: pbp.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.3 Spectrum Fitting Through χ2 Minimization: lma.f
vii
. . . . . . . . . . 197
List of Tables
1.1
Basic physical properties of main sequence stars of various spectral types.
2.1
Log of observations of the HD 72106 system obtained with ESPaDOnS. 47
3.1
Fundamental physical parameters for HD 72106A and B. . . . . . . .
3.2
Mean longitudinal field measurements for each observation of HD 72106A
and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
3
81
96
Chemical abundances for HD 72106A in each independently fit segment
of spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2
Averaged best fit chemical abundances, v sin i and microturbulence for
HD 72106A and B as well as solar abundances. . . . . . . . . . . . . . 118
4.3
Chemical abundances for HD 72106B in each independently fit segment
of spectrum.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
viii
List of Figures
1.1
Sample H-R diagram illustrating main sequence stars as well as white
dwarfs, giants and supergiants.
1.2
. . . . . . . . . . . . . . . . . . . . .
Typical chemical abundances of an Ap star (HD 162576) and a ‘normal’
A star (HD 162817). . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
.
10
Pre-main sequence evolutionary tracks from Palla & Stahler (1993) for
a range of masses.
2.1
8
Variability of the longitudinal magnetic field, He line intensity and
u-band magnitude in the star HD 184927, from Wade et al. (1997)
1.4
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Schematic diagram of the combinations of polarization states used to
calculate the Stokes parameters.
. . . . . . . . . . . . . . . . . . . .
34
2.2
Illustration of the ‘normal’ Zeeman effect. . . . . . . . . . . . . . . .
36
2.3
An observation of line splitting in HD 94660 due to the Zeeman effect.
39
2.4
Schematic of the polarization analysing optics in ESPaDOnS. . . . . .
43
2.5
Schematic of the ESPaDOnS spectrograph. . . . . . . . . . . . . . . .
45
2.6
Sample flat field and thorium argon calibration frames. . . . . . . . .
48
2.7
Unnormalized, with a polynomial fit, and normalized spectra. . . . .
55
2.8
Sample Hβ and metallic lines in observations of HD 72106B on its own
illustrating the stability of the spectrum. . . . . . . . . . . . . . . . .
ix
59
2.9
Sample Hβ and metallic lines in observations of HD 72106A on its own
illustrating the stability of the spectrum. . . . . . . . . . . . . . . . .
60
2.10 Sample reconstructed spectra of the secondary, in a stable region of
the primary’s spectrum. . . . . . . . . . . . . . . . . . . . . . . . . .
61
2.11 Variable metallic lines in observations of HD 72106A on its own. . . .
63
2.12 Comparison of a spectrum of the primary, combined light from both
components and a reconstructed primary spectrum. . . . . . . . . . .
65
2.13 Reconstructed spectra of HD 72106A at different phases, illustrating
variability as well as the accuracy of the reconstruction technique. . .
66
3.1
Sample Balmer line fits of Hγ for HD 72106A & B. . . . . . . . . . .
70
3.2
Sample best fit spectra for HD 72106B at both Teff = 8000 K, log g =
4.5 and Teff = 8750 K, log g = 4.0. . . . . . . . . . . . . . . . . . . .
3.3
71
H-R diagram containing HD 72106A and B along with pre-main sequence evolutionary tracks and isochrones.
. . . . . . . . . . . . . .
75
3.4
An illustration of orbital geometries discussed in Section 3.2.
. . . .
78
3.5
Sample best fit synthetic spectra for HD 72106A. . . . . . . . . . . .
87
3.6
Sample best fit synthetic spectra for HD 72106B. . . . . . . . . . . .
88
3.7
Sample LSD profiles for HD 72106A & B in both Stokes I and V. . .
93
3.8
Sample χ2 map in Bp and β for θ 1 Ori C, adapted from Wade et al.
(2006a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9
99
Periodogram for HD 72106A based on longitudinal field measurements,
created using a first-order sinusoid. . . . . . . . . . . . . . . . . . . . 103
3.10 Periodograms for HD 72106A based on the LSD Stokes I and V profile
variations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
x
3.11 Illustration of the rotational Doppler shift for Doppler Imaging . . . . 108
4.1
Additional best fit spectra for HD 72106A in two windows.
. . . . . 117
4.2
Abundances relative to solar for HD 72106A and B, averaged over all
spectral windows modeled. . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3
Additional best fit spectra for HD 72106B in two windows.
4.4
Emission and variability in the Hα Balmer line of HD 72106B. . . . . 124
4.5
Emission and variability in the OI 7773 Å triplet of HD 72106B.
4.6
Longitudinal magnetic field measurements of HD 72106A, phased with
the adopted 0.63995 day period, and the best fit sinusoid.
4.7
. . . . . 123
. . 125
. . . . . . 128
Phased LSD profiles for HD 72106A with the 0.63995 day best fit
period.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.8
Phased LSD profiles for HD 72106A with the inferior 1.7859 day period. 131
4.9
Phased LSD profiles for HD 72106A with the inferior 1.6921 day period. 132
4.10 Phased LSD profiles for HD 72106A with the inferior 0.38983 day period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.11 Maps of reduced χ2 for a range of dipole field models at different inclination angles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.12 Synthetic Stokes V LSD profiles for the best fit magnetic field geometry
compared with the observed profiles. . . . . . . . . . . . . . . . . . . 137
4.13 Doppler mapping reconstruction of Si for HD 72106A from original
lines.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.14 Line fits for individual Si lines used in Doppler mapping of HD 72106A. 140
4.15 Doppler mapping reconstructions of Si, Ti, Cr, and Fe for HD 72106A
from LSD profiles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
xi
4.16 LSD line profile fits from Doppler Imaging of Si, Ti, Cr, and Fe for HD
72106A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
xii
Chapter 1
Introduction
1.1
The Diversity of Stars
Stars exhibit a tremendous range of evolutionary and physical properties. They run
from pre-main sequence stars that are still forming and have only just become visible
to optical telescopes, to white dwarfs and neutron stars that are truly ancient, even by
astronomical standards. Stellar masses can range from a tenth of the mass of the Sun
to 100 times the Sun’s mass (M ). Radii of stars can approach 1000 times the Sun’s
radius (R ) in the case of supergiants, and dwindle to the scale of 10 kilometers in the
case of ultra-dense neutron stars. Many of these objects are exotic, rare, and short
lived. A casual examination of the variety of known stars suggests a bewildering,
and potentially incomprehensible, diversity. However, if one restricts oneself to main
sequence stars, a somewhat clearer picture emerges. Main sequence stars are stars
in their ‘prime’; objects which are long lived, in hydrostatic equilibrium and which
derive their energy from fusion of hydrogen into helium deep in their cores. These
stable stars at first glance seem like the simplest objects to investigate. However,
1
CHAPTER 1. INTRODUCTION
2
upon closer inspection a number of complications arise.
The main sequence is defined by stars that differ primarily in age and mass. These
stars occupy a diagonal strip along the Hertzsprung Russell (H-R) diagram, a plot
of stellar surface luminosity versus temperature (see Figure 1.1). Differences in age
and mass translate into differences in luminosity and temperature, thus defining a
continuum of stars. The H-R diagram can be a very useful tool for examining large
scale stellar properties. In particular, if one can create accurate models of the large
scale properties of stars, one can compare a star’s position on the H-R diagram with
the points generated by models. Thus one can say something about a star’s mass and
evolutionary state based on its observed surface temperature and luminosity.
Figure 1.1: A typical H-R diagram illustrating main sequence stars, as well as white
dwarfs, giants and supergiants. Luminosity is in units of solar luminosities. Image:
M. Fanelli, Texas Christian University, http://personal.tcu.edu/∼mfanelli/
Stars of different temperatures (and masses) are broadly grouped into different
spectral classes, according to the characteristics of their spectra. These classes are
3
CHAPTER 1. INTRODUCTION
Spectral Type
O
B
A
F
G
K
M
Teff (K)
30, 000
10, 000
7, 500
6, 000
5, 000
3, 500
2, 000
-
60, 000
30, 000
10, 000
7, 500
6, 000
5, 000
3, 500
Log L/L
Mass (M )
Radius (R )
6.14
4.30
1.90
0.77
0.07
−0.39
−1.39
60
18
3.1
1.7
1.1
0.8
0.3
15
7
2.1
1.3
1.1
0.9
0.4
Table 1.1: Basic physical properties of main sequence stars of various spectral types.
Logarithmic luminosity, mass and radius are given in solar units, and represent upper
bounds for the spectral class (adapted from Ostlie & Carroll, 1996).
labeled, from hottest to coolest, by the letters O, B, A, F, G, K, and M. Historically these classifications were based on the strengths of the Balmer absorption lines,
Helium absorption lines, and a few other metallic lines. Recent discoveries of very
cool brown dwarfs have led to the addition of, arguably non-canonical, classes L and
T (Kirkpatrick, 2005). Sub-classes numbered 1 through 9, decreasing with temperature, are also defined. For example, our Sun is a main sequence star, with a G2
spectral type (Ostlie & Carroll, 1996). Hotter stars are often referred to as ‘early
type’ stars, while cooler stars are referred to as ‘late type’. A brief overview of the
physical properties of different stellar spectral types is provided in Table 1.1.
Stars are also broadly classified according to their luminosity, based on the widths
of pressure broadened spectral lines. These luminosity classes are defined by Roman
numerals ranging from I at the most luminous to V at the least. Main sequence stars
are of luminosity class V; giants are class III, and super giants are class I (Ostlie &
Carroll, 1996).
Observations of stars are obtained, almost exclusively, of the star’s photosphere.
The photosphere is the name given to the outer layers of a star, which provide the
4
CHAPTER 1. INTRODUCTION
transition from the dense stellar interior to the vacuum of space. The photosphere
is the region of the star in which the opacity becomes sufficiently low for photons
to escape and eventually reach our telescopes. Thus, this is the region probed by
photometric and spectroscopic measurements, and the region in which observable
spectral lines are formed.
Two important definitions to provide before further discussion of stellar astrophysics are effective temperature (Teff ) and surface gravity (log g). Effective temperature is the temperature a star would have, given its total luminosity integrated over
all wavelengths, if it were a perfect spherical black body. That is:
2
L = 4πR2 σTeff
,
(1.1)
where L is the stars’ luminosity, R is the stellar radius and σ is the Stefan-Boltzmann
constant (Ostlie & Carroll, 1996). This is, of course, just the Stefan-Boltzmann equation for a uniform spherical emitter. The surface gravity, g, is simply the acceleration
due to gravity at the surface of the star. In logarithmic units it is given by:
log g = log GM − 2 log R,
(1.2)
where M is the mass of the star and G is the gravitational constant.
While this presents an apparently straightforward view of main sequence stars,
the devil is in the details. A variety of physical processes can significantly modify
the observable properties of stars, as well as stellar structure and evolution. Many
of these properties are not included, or are included only approximately, in current
stellar models. For example, in the case of massive O stars, mass loss due to strong
stellar winds can complicate the picture (Massey, 2003), particularly when coupled
with magnetic fields (Donati et al., 2002). In the case of very cool stars, departures
CHAPTER 1. INTRODUCTION
5
from local thermodynamic equilibrium (Asplund, 2005) and molecular bands in stellar
spectra (e.g. Cushing et al., 2005) are poorly understood. Even in the case of our Sun,
the internal rotation structure and details of how the solar magnetohydrodynamic
dynamo generates the observed magnetic field are not clear (Thompson et al., 2003).
These particular problems are minimized for main sequence A and B stars, the focus
of this thesis. However, a number of interacting effects of comparable magnitude, such
as atomic diffusion and magnetic fields discussed below, lead to other complications
in these stars.
1.1.1
Main Sequence A and B Stars
Main sequence A and B stars are stars in the temperature range 7500 K to 30000 K
(see Table 1.1). They have masses from about 1.6M to 17M and, as such, are often
referred to as intermediate mass stars. The structure of these stars consists qualitatively of a convective core (which increases in volume with increasing stellar mass)
with a larger overlying radiative envelope. That is to say, in the radiative envelope
the star is stable against strong convection, and energy is transported primarily via
radiation. In the case of cooler A-type stars, energetically weak convection can occur
(Vauclair & Vauclair, 1982; Landstreet, 1998). The photospheres of A and B stars are
generally in local thermodynamic equilibrium (LTE)1 . Optical spectra of these stars
display very strong Balmer absorption lines, with a wide variety of metallic2 absorption lines. In the large majority of A and B stars, line emission is not observed, nor
1
Local thermodynamic equilibrium is defined by a negligible temperature change over the mean
free path of a photon. In general, LTE models of A and B stars reproduce detailed observations of
these stars very reliably. However, some exceptions exist, for example emission lines in spectra of
Herbig Ae/Be stars.
2
In the astronomical community the terms ‘metals’ or ‘metallic’ refer to elements heavier than
He.
CHAPTER 1. INTRODUCTION
6
is there any evidence for magnetic fields or magnetic activity (Borra et al., 1982).
1.1.2
Chemically Peculiar A and B Stars
In the early days of stellar spectroscopy it was noticed that a small number of A and
B stars displayed anomalous metal line strengths in their spectra. Antonia Maury is
credited with the discovery of these stars in 1897 (Maury & Pickering, 1897). She
encountered them as part of a larger classification effort, and flagged them as ‘peculiar’. This historical designation has been retained in the modern label ‘chemically
peculiar stars’.
Chemically peculiar stars display abundances of some elements that depart substantially from the abundances seen in the Sun, and in other main sequence stars of
similar type. These abundance anomalies vary from element to element, and some
elements may have nearly solar abundances. These abundance anomalies are not a
result of nucleosynthetic processes in the star, or an anomalous chemical mixture of
the material from which they have formed (Michaud, 1970; Landstreet, 1992). In fact,
it is not uncommon to find chemically peculiar stars closely associated with ‘normal’
A and B stars in binary systems or open clusters (e.g. Silvester, 2007), where they
must have shared a very similar formation and evolution. Rather, peculiarities are
widely believed to be a superficial effect, resulting from chemical fractionation within
the outer layers of a star. In a sufficiently stable stellar atmosphere, the competing effects of radiation pressure and gravity allow for the diffusion of elements into regions
of over- or under-abundance (Michaud, 1970), producing the observed abundance
anomalies.
CHAPTER 1. INTRODUCTION
7
Ap and Bp stars
Overall, chemically peculiar stars make up only about 10% of main-sequence A and
B stars (Hoffleit & Jaschek, 1991). These stars can be organized into various groups
based on the types of chemical peculiarity observed. The Ap and Bp star group
is one of the more interesting. These stars are uniquely characterized by strong
globally-ordered magnetic fields, which can usually be well approximated by a low
order multipole (e.g. a dipole). Magnetic fields in Ap and Bp stars were first observed
by Babcock (1947). The field strengths at the surface of the star range from several
hundred gauss (Aurière at al., 2007) to greater than 10 kilogauss (Bagnulo et al., 2003,
2004). These properties stand in remarkable contrast to the magnetic fields observed
in the sun and other low-mass stars. In these low-mass stars local spots with strong
magnetic fields, on the order of kilogauss, are seen, but the global average magnetic
field is near zero (e.g. Thompson et al., 2003). Over-abundances3 of iron peak elements, by up to 100 times the respective solar abundance, are sometimes seen in Ap
and Bp stars. Over-abundances of silicon ranging above 10 times the solar abundance,
and some rare earth elements, sometimes (apparently) exceeding 1000 times the solar
abundance, are also seen in these stars (e.g. Folsom et al., 2007). Under-abundances
of some light elements, particularly He, are often observed. Chemical abundances of
a typical Ap star and a typical A star are compared in Figure 1.2. In addition to
chemical abundance anomalies, Ap and Bp (often abbreviated to Ap/Bp, or simply
Ap) stars display inhomogeneities in chemical abundance across the surface of the
star, essentially large-scale chemical star spots (Rice et al., 1989; Hatzes et al., 1989).
All of these properties of Ap/Bp stars are stable over long periods of time; to
3
Over-abundances and under-abundances are always in reference to solar abundances in this
thesis.
CHAPTER 1. INTRODUCTION
3
8
HD 162725
Abundance
2
1
0
-1
-2
3
HD 162817
Abundance
2
1
0
-1
-2
C O Ne Mg Si Ca Sc Ti V Cr Mn Fe Co Ni Ba La Ce Pr Nd Sm Eu
Figure 1.2: Typical chemical abundances of an Ap star, HD 162576, and a ‘normal’
A star, HD 162817, both in in the open cluster NGC 6475 (from Folsom et al., 2007).
Abundances are presented in logarithmic units relative to solar abundances. Downward arrows represent upper limits and the dashed line represents solar abundance
values. In the peculiar star HD 162725 (the top frame) strong over-abundances of
Cr, Mn, Fe, and some rare earth elements are inferred. O and Mg appear to be
under-abundant in this star. In contrast, in HD 162817 these abundances are all
approximately solar.
CHAPTER 1. INTRODUCTION
9
date no observations have indicated any evolution of the magnetic field, chemical
abundances or surface inhomogeneities of individual stars. Periodic variability is
commonly observed, particularly in photometric brightness, the observed longitudinal
(line-of-sight) magnetic field, and absorption line shapes. This is due to modulation
caused by the rotation of the star rather than any true variation in the underlying
properties. These periodic variations are generally described as an ‘oblique rotator’
(Stibbs, 1950). In this scenario, a property, for example magnetic field, is distributed
non-uniformly over the entire stellar surface. As the star rotates, different parts of
the star become visible. Hence, different disk-averaged properties are observed. Such
variations can extend even to photometric variability of the star. If an over-abundant
spot of an element that contains many absorption lines rotates into view, the total
amount of light emitted by the star in a particular wave band can decrease due to ‘line
blanketing’ (e.g. Wade et al., 1997; Ryabchikova, et al., 1997). Figure 1.3 illustrates
variability in several properties of the Bp star HD 184927 as the star rotates, due to
non-uniform magnetic and chemical surface distributions.
Ap stars are loosely classified into several subtypes, based on some of their stronger
spectral peculiarities. Most commonly, a distinction is made between SrCrEu stars
and Si stars. SrCrEu stars are cooler (6500-10000 K), generally A-type and early
F-type stars, while Si stars are hotter (10000-15000 K), late to mid-B-type stars.
One particularly interesting subgroup is the rapidly oscillating Ap stars (roAp stars).
These stars pulsate non-radially with periods on the order of minutes (Kurtz, 1990),
and the excitation mechanism is still poorly understood (Kochukhov et al., 2007a).
One additional oddity seen in magnetic Ap/Bp stars is that as a group, they
rotate much more slowly than normal A and B stars. Generally they have rotation
CHAPTER 1. INTRODUCTION
10
Figure 1.3: Variability in longitudinal magnetic field (in gauss), He line intensity (in
arbitrary units, 1 = normal line strength, 5 = very strong) and u-band magnitude
versus phase of rotation in the Bp star HD 184927, from Wade et al. (1997). This
provides an illustration of the observed properties of an oblique rotator, with an
inhomogeneous surface chemical abundance distribution and magnetic field.
CHAPTER 1. INTRODUCTION
11
velocities, projected along the line of sight, (v sin i) of around 60 km s−1 or less,
whereas ‘normal’ A and B stars usually have v sin i above 120 km s−1 (Abt & Morrell,
1995). Directly measured rotation periods of Ap/Bp stars generally are between 1 to
10 days (Power, 2007). Some Ap stars are sufficiently slowly rotating that they exhibit
no measurable rotational broadening of their spectral lines, even in extremely high
resolution (R ∼120000) spectra. These objects often have poorly constrained, but
very long (sometimes on the order of decades), rotation periods (Stȩpień & Landstreet,
2002; Power, 2007).
A number of enduring questions exist about Ap/Bp stars, particularly regarding
the origin of the magnetic field and the detailed physical processes chat produce and
maintain the chemical peculiarities. These will be addressed somewhat further in
Sections 1.2 and 1.3.
Other Chemically Peculiar Stars
A number of other classes of chemically peculiar stars exist. The He weak stars
(sometimes called He weak SiSrTi stars), essentially a continuation of the Bp Si stars
to higher Teff and M , show strong under-abundances of He. These stars display
over-abundances of a number of iron peak elements as well as silicon, and possess
magnetic fields. He weak stars are of intermediate B-type, in the Teff range 1200019000 K (Cidale et al., 2007).
The group of He strong stars also display magnetic fields and abundance anomalies, most prominently an overabundance of He. These stars appear to be an extension
of the He weak group, with temperatures in the 19000-25000 K range (Cidale et al.,
2007).
CHAPTER 1. INTRODUCTION
12
There also exist a number of non-magnetic classes of chemically peculiar stars. In
the cooler range, there are the Am stars. These so called metallic-line stars display
over-abundances of heavier elements, particularly rare earths, and under-abundances
of lighter elements. They rotate slowly (Abt & Morrell, 1995) and inhabit the 700010000 K temperature range (Kurtz & Martinez, 2000).
In approximately the same temperature range there exists the rare class of λ Boötis
stars. These are non-magnetic stars (Bohlender & Landstreet, 1990) that display
solar abundances of lighter elements, but under-abundances of iron peak elements.
Non-radial pulsations have been observed in these objects (Bohlender et al., 1999).
Mercury-manganese (HgMn) stars, as the name suggests, display over-abundances
of Hg and Mn, and are another class of non-magnetic chemically peculiar star (e.g.
Shorlin et al., 2002). They have temperatures in the 10000-15000 K range (Kurtz
& Martinez, 2000). These stars have become a subject of great interest in the last
few years as they are the only non-magnetic class of star known to display chemical
abundance star spots on their surfaces (Adelman et al., 2002; Wade et al., 2006b).
Additionally, recent observations suggest these spots change on timescales of years
(Kochukhov et al., 2007b), which is truly surprising.
The non-magnetic stars known as He weak PGa stars are characterized by overabundances of phosphorus and gallium and under-abundances of helium. These nonmagnetic objects are essentially higher temperature analogs of HgMn stars, with
temperatures in the 13000-18000 K range (Kurtz & Martinez, 2000).
CHAPTER 1. INTRODUCTION
1.2
13
Diffusion
The wide range of chemical peculiarities seen in A and B stars are attributed to the
effects of atomic diffusion, as described in the seminal paper by Michaud (1970). Diffusion occurs under the competing effects of gravity and radiation pressure. Under
gravity alone, elements heavier than H would tend to sink in a stable star. However,
the slightly anisotropic radiation field in the star provides a net outward radiative
force that can potentially counter gravity, raising atoms up in the stellar atmosphere.
A levitated atom of a particular species will tend to rise in the atmosphere until combined changes in the radiation field, atomic excitation and atomic ionization balance
the effect of gravity, yielding a net acceleration which is null. The element will tend
to accumulate in that region producing a local over-abundant layer. Thus atoms of
various elements become concentrated in layers at different heights in the atmosphere,
depending on the particular atomic properties of the element and physical properties
of the star. When the star is examined spectroscopically, these layers are observed as
general over-abundances or under-abundances if they occur in the photosphere. If an
over-abundant layer of an element is located near the surface of a star, in the region in
which the stellar spectrum is formed (the photosphere), then more light is absorbed
by that element, and the corresponding absorption lines are seen to be stronger, and
hence an over-abundance is deduced. Conversely, if a layer of local under-abundance
of an element is near the surface of the star (e.g. if the net force on the element is
negative and the element sinks in the photosphere), then the absorption lines formed
are weaker, and hence an under-abundance is deduced.
For an atom to experience radiation pressure it must absorb photons, either
through bound-bound or through bound-free transitions. Following Michaud (1970),
CHAPTER 1. INTRODUCTION
14
bound-free transitions can only occur frequently for atoms with an ionization potential less than 13.6 eV. This is the ionization potential of hydrogen, and photons above
this energy are almost all absorbed by neutral H. Hence there is not enough flux above
this energy to levitate atoms. Atoms with an ionization potential less than 10 eV
will not acquire enough momentum from the radiation field to counter gravity, and
will sink into the star4 . Thus only atoms (or ions) with ionization potentials in the
10–13.6 eV range can be pushed up in the stellar atmosphere by bound-free radiation
pressure. Species with ionization potentials in this range include C I, Ca II, Sr II,
and most (singly ionized) rare earth elements. Atoms outside of this range may still
absorb enough radiation in bound-bound transitions to counter gravity. Species with
many absorption lines, particularly rare earth elements and some iron peak elements,
can form layers of substantial over-abundance through this mechanism.
Michaud (1970) showed that this vertical diffusion process can take place on the
order of 104 years, much shorter than the lifetime of a star. Thus, the time scale for
the development of abundance anomalies by diffusion is compatible with the presence
of peculiarities in the youngest intermediate mass main sequence stars, with ages of
order ∼ 106 years.
A critical criterion for atomic diffusion is the stability of the stellar atmosphere.
Turbulence can easily overwhelm atomic diffusion, providing a homogeneous atmosphere. Turbulence tends to move material about in the stellar atmosphere, homogenizing it. Thus, for diffusion to play an important role, turbulent velocities in a
star must be less than the relatively small diffusion velocities. Michaud (1970) found
diffusion velocities on the order of 10−3 cm sec−1 . Intermediate mass A and B-type
4
These models assume an A or B type stellar atmosphere, specifically Teff = 12600 K and
log g = 4.0. Michaud (1970) examined a range of temperatures from 8000 K to 20000 K and found
largely the same trends in all models.
CHAPTER 1. INTRODUCTION
15
stars possess radiative atmospheres, which have little turbulence and virtually no convection, allowing diffusion to potentially build up large chemical peculiarities. Lower
mass stars, such as our Sun, have deep convective atmospheres. Although diffusion
may occur in these lower mass stars, the deep mixing of their atmospheres homogenizes any resultant peculiarities to undetectable levels. This provides an explanation
for why chemical peculiarities are not seen in these cooler classes of stars. O and early
B-type stars possess large convective cores which can approach the surface of the star
which, combined with the overshoot of convective cells into the radiative envelope,
can disrupt the atmosphere of the star. More importantly, O and early B-type stars
display significant mass loss via high-velocity stellar winds, which can lead to mixing
in the stellar atmosphere, preventing the build up of chemical peculiarities (Vauclair
& Vauclair, 1982).
Mixing due to rotation can also compete with diffusion. Rotation produces meridional circulation, which can vertically mix a star’s atmosphere. In the case of a normal
A or B star with an equatorial rotation velocity of ∼ 100 km s−1 , meridional circulation can produce velocities more than sufficient to disrupt diffusion (Michaud, 1982).
However, for more slowly rotating stars (and in particular virtually all chemically
peculiar stars), meridional circulation usually produces velocities less than the diffusion velocity, allowing chemical stratification to occur. This typically requires an
equatorial velocity of less than 90 km s−1 (Michaud, 1982).
In the case of magnetic chemically peculiar stars, the magnetic field can play
several important roles in the production and maintenance of chemical abundance
structures. The field can help to stabilize the atmosphere of the star (Michaud,
1970), in particular by suppressing horizontal mixing. The magnetic field can also
CHAPTER 1. INTRODUCTION
16
strongly constrain atoms to follow the field lines (Michaud et al., 1981; Alecian &
Stift, 2006). This may provide a mechanism for the development of surface abundance
inhomogeneities, either by direct magnetohydrodynamic modification of the diffusion
velocity, or by influencing other phenomena, such as a weak stellar wind. Additionally,
the Zeeman splitting of atomic lines in the presence of a magnetic field can increase
the opacity of some species, and hence increase the radiation pressure they experience
(Alecian & Stift, 2002, 2006). This can further increase the efficiency of diffusion for
some elements.
In the presence of a weak stellar wind, elements that might otherwise accumulate
at the surface of a star can be ejected from the star entirely. This can cause a selective
depletion of some elements (Babel, 1992). On the other hand, such a wind might drag
some elements towards the surface of the star, depositing them there if they do not
experience sufficient force to be ejected. Additionally, the presence of a magnetic field
can affect the wind geometry, and hence modify the wind’s effects.
As this overview suggests, atomic diffusion in intermediate mass stars is a rather
complex mechanism, involving a large variety of competing processes. The details of
these processes are generally understood only approximately (Alecian & Stift, 2006).
Diffusion has a wide range of impacts on the observed properties and physical structure of stellar photospheres, and is critical to the understanding of chemically peculiar
stars. Consequently, understanding the observable results of diffusion provides valuable information about poorly understood stellar physics, including rotational mixing,
convection, stellar winds, and magnetic fields.
CHAPTER 1. INTRODUCTION
1.3
17
Magnetism in Intermediate Mass Stars
The first direct observation of a magnetic field in a star (besides the Sun) was obtained
by Babcock (1947). In this study he used the longitudinal Zeeman effect to detect a
magnetic field in the Ap star 78 Vir.
The study of stellar magnetism relies heavily on the use of astrophysical spectropolarimetry. A spectropolarimeter is essentially the combination of a spectrograph and
a polarimeter. The polarimeter analyses light and splits it into the desired polarization states, then the analyzed light is dispersed and recorded by the spectrograph.
In the Zeeman effect, one atomic absorption (or emission) line is split into multiple
components in the presence of a magnetic field, with the spacing between the components being linearly proportional to the strength of the field. Additionally, different
components of the split line have different polarization properties. The Zeeman effect is discussed further in Section 2.1.2. In a line profile broadened by the Doppler
effect due to stellar rotation, the polarization properties of the Zeeman effect are
much easier to observe than the wavelength splitting of the line in unpolarized light.
Thus, with a spectropolarimeter, one can observe the Zeeman effect in the absorption
lines of a star with high precision, and hence determine the properties of the stellar
magnetic field.
Although observations of magnetic fields in Ap/Bp stars are now obtained routinely, the source of these magnetic fields remains a mystery. There are currently
two broad classes of theories for the origin of the observed magnetic fields. The fossil
field theory suggests that the observed magnetic field is a remnant from an earlier
stage of stellar evolution, possibly dating back to the original gas cloud from which
the star formed, that has since become ‘frozen’ into the star. Current models suggest
CHAPTER 1. INTRODUCTION
18
that fossil magnetic field configurations, which are stable over the lifetime of a star,
do exist (Braithwaite & Nordlund, 2006). However, it is not clear if a magnetic field
from the interstellar medium could survive the possibly violent evolution along the
Hayashi track to the main sequence (mixing could cause very complex magnetic field
geometries, which would be unobservable and decay quickly), or if a temporary dynamo could possibly generate an appropriate field during pre-main sequence evolution
(Moss, 2003).
The contemporaneous dynamo theory, currently less favored than the fossil field
theory, suggests that the magnetic field is generated contemporaneously by a dynamo,
probably deep in the star’s convective core. The field then must diffuse through the
large radiative envelope to emerge at the surface of the star to become visible. It is not
clear if such a dynamo generated field could produce the roughly dipolar, misaligned
surface field geometries seen, or if the magnetic field could rise to the surface in
a sufficiently short time to produce fields in the youngest observed magnetic stars
(Moss, 2001; Charbonneau & MacGregor, 2001). Additionally, there is a lack of
any significant correlation between field strength and rotation rate (Kochukhov &
Bagnulo, 2006; Landstreet et al., 2007), unlike the strong correlation observed in
solar-type dynamos. The ability of very slow rotating stars to possess (often very
strong) magnetic fields, and the lack of a rotation period-field strength relation, casts
further doubt on contemporaneous dynamo models.
In this context, the study of magnetic fields in intermediate mass pre-main sequence stars becomes particularly interesting. These stars can provide strong constraints on the origin of the observed magnetic fields in Ap/Bp stars. The presence of
strong, globally-ordered magnetic fields in pre-main sequence stars is almost certainly
CHAPTER 1. INTRODUCTION
19
necessary for the fossil field hypothesis to be true. On the other hand, detections of
magnetic fields in pre-main sequence stars place strong constraints on contemporaneous dynamo models. In particular, such detections require that the magnetic field
diffuses through the stellar atmosphere on a time-scale shorter than the pre-main
sequence lifetime of the star, quite a bit faster than the constraints obtained from the
main-sequence. Additionally, such observations require that a dynamo, if it exists,
‘turns on’ early in the star’s evolution, and then persists through the pre-main sequence and the main sequence. Conversely, it may be possible to observe a pre-main
sequence dynamo that ‘turns off’, leaving behind a fossil field. Furthermore, the internal structure of pre-main sequence differs significantly from that of main sequence
stars, and hence these young stars can show how stellar magnetic fields behave in
different conditions. Consequently, a detailed understanding of magnetic fields in intermediate mass pre-main sequence stars is of great importance. This provides one
of the major motivations for the study of the HD 72106 system.
1.4
Herbig Ae and Be stars
Herbig Ae and Be stars (abbreviated as HAeBe stars) are pre-main sequence stars of
intermediate mass and spectral types A and B. These objects were first identified as
a stellar class by Herbig (1960). HAeBe stars evolve into A and B stars when they
reach the main sequence. As such, they are higher mass analogs of the better known,
lower mass, T Tauri stars.
Identifying HAeBe stars can be challenging, and there has been much discussion of
the exact criteria for classification in the literature (e.g. Thé et al., 1994). Historically
HAeBe stars have been identified as spectral type A or B stars with optical emission
CHAPTER 1. INTRODUCTION
20
lines, located in an obscured region, and associated with bright nebulosity (Herbig,
1960). These criteria have since been refined and supplemented. The most widely
accepted criteria (Vieira et al., 2003) are currently:
1. Spectral type of A or earlier, with emission lines
2. Located in an obscured region
3. Fairly bright nebulosity nearby
4. Possess an anomalous extinction law
5. Display infrared excess
6. Are photometrically variable
7. Show emission in the Mg II λ 2800 Å line.
Very few HAeBe stars satisfy all of these criteria; generally classification of HAeBe
stars only requires that several of these criteria be met. The primary objective of
the selection criteria chosen is to distinguish between pre-main sequence HAeBe stars
and main sequence or post-main sequence objects with circumstellar material (e.g.
classical Be stars or AGB stars).
1.4.1
Observed Properties
The spectral energy distributions (SED) of HAeBe stars tend to display clear effects
of circumstellar matter. Thermal emission by circumstellar dust can dominate the
SED in the infrared, providing a large excess of flux. In the ultraviolet, circumstellar
material can also sometimes dominate the SED. This is attributed to hot gas near
CHAPTER 1. INTRODUCTION
21
the surface of the star, exceeding the star’s temperature (Meeus et al., 1998). This
increase in temperature can be interpreted as heating due to accretion, implying
an accretion rate on the order of 10−7 M year−1 (Blondel & Tjin A Djie, 1994).
It is important to note that the lack of veiling (emission or absorption from the hot
boundary layer ‘in front’ of the star) in optical spectra indicates that the contribution
from this hot gas at visible wavelengths is small for most stars (Böhm & Catala, 1993;
Ghandour et al., 1994).
X-ray emission has been reported for a number of HAeBe stars (Zinnecker &
Preibisch, 1994; Damiani et al., 1994). However, despite steadily improving observations, the source of these X-rays remains a mystery. The X-rays have been attributed
to unresolved T Tauri companions (Stelzer et al., 2006), which emit X-rays in a fashion similar to the Sun. However, magnetically confined winds around the HAeBe star
itself have also been used to explain this emission (Telleschi, 2007). Circumstantial
supporting evidence for both hypotheses exists, but no consensus has been reached.
Photometric variability is observed in HAeBe stars, likely due to variations in
extinction from circumstellar material (e.g. van den Ancker et al., 1998). Linear
polarization (in the continuum) is also observed, attributed to scattering from a nonspherical distribution of dust (Grinin, 1994). Variations in the polarization are seen,
often correlating with photometric variations (Grinin et al., 1994; Waters & Waelkens,
1998).
Infrared spectroscopy has been a subject of some interest, as it provides an avenue
for examining the geometry and composition of circumstellar material around HAeBe
stars. Absorption lines from a number of molecules have been observed. Species
observed include: FeO, polycyclic aromatic hydrocarbons (PAHs), various amorphous
CHAPTER 1. INTRODUCTION
22
and crystalline silicates, and H2 O ice (e.g. Waters & Waelkens, 1998; Schütz et al.,
2005).
Spectroscopy in the optical spectral region reveals spectra that bear similarities to
those of main sequence A and B stars. These spectra mostly probe the photosphere
of HAeBe stars, and are therefore similar to spectra of main sequence stars, outside of
the emission lines. Moderate projected rotation velocities (v sin i) in the range 60-200
km s−1 are observed (Böhm & Catala, 1995). Photospheric chemical abundances are
seen to be roughly solar in HAeBe stars (Dunkin et al., 1997; Acke & Waelkens, 2004).
However, in some cases hints at departures from solar abundance have been observed
(Beskrovnaya, et al., 1999); for example, a general depletion of elements relative to
solar (Acke & Waelkens, 2004). Emission is seen in the Balmer series, particularly
Hα, as well as a number of other ions; commonly O I, Ca II, Si II, Mg II, and Fe II.
The source of these emission lines is a subject of some debate. For example, Hamann
& Persson (1992) favor a model in which emission lines are formed in the turbulent
region where accreting matter meets the stellar surface, whereas Catala et al. (1999)
argue that the emission lines are formed due to a chromospheric wind. The emission
lines, and in some rare cases apparently absorption lines, display complex variability
(Catala et al., 1999).
Recently magnetic fields have been detected in some HAeBe stars (Donati et al.,
1997; Hubrig et al., 2004; Wade et al., 2005, 2007). Wade et al. (2005, 2007) argue that
these fields are globally ordered and have longitudinal strengths of several hundred
gauss. This is particularly interesting given the questions surrounding the origin of
magnetic fields seen in Ap/Bp stars. Because HAeBe stars evolve into A and B
stars, it may well be that the observed magnetic HAeBe stars evolve into magnetic
CHAPTER 1. INTRODUCTION
23
Ap/Bp stars. Supporting this hypothesis, the observed magnetic field strengths and
geometries are approximately the same in magnetic HAeBe stars and Ap/Bp stars
(Wade et al., 2005; Catala et al., 2007; Wade et al., 2007; Alecian et al., 2007, in
preparation). Furthermore, the observed incidence of magnetic HAeBe stars in the
set of all HAeBe stars is about the same as the incidence of Ap/Bp stars in all A
and B stars (Wade et al., 2005, 2007). The observation of magnetic fields in HAeBe
stars will have broader implications for models of HAeBe star X-ray emission based
on magnetically confined winds, and for models of magnetically mediated accretion.
However, further study is necessary before any conclusions can be drawn.
1.4.2
Physical Properties
The geometry of the circumstellar material around HAeBe stars, not surprisingly,
remains somewhat controversial. It now appears that there is strong evidence for the
presence of massive accretion disks around many HAeBe stars (Marsh et al., 1995;
Eisner et al., 2004; Corder et al., 2005). These disks have radii on the order of several
hundred astronomical units (AU), and masses in the range 0.005 M to 0.05 M .
The details of the disk geometries continue to be the subject of investigation (Eisner
et al., 2004; Corder et al., 2005). Strong evidence for larger, more diffuse envelopes
of gas around a number of HAeBe stars is also reported (Natta et al., 1993; van der
Tak et al., 2000), adding another element to the picture.
In evolutionary terms, HAeBe stars are objects that have recently become observable in visible light and are evolving towards the main sequence. These stars form
the middle ground in pre-main sequence objects; falling between the lower mass T
Tauri stars and the (largely unobserved) higher mass pre-main sequence stars. They
CHAPTER 1. INTRODUCTION
24
are found in the mass range ∼ 2 M to ∼ 8 M (Palla & Stahler, 1993). Stars more
massive than 8 M do not have an optically observable pre-main sequence life. These
stars have already commenced hydrogen burning when their protostellar accretion
phase ends and they become observable. Stars below 2 M are radiatively unstable,
largely convective, and hence evolve as T Tauri stars. However, some authors include
stars down to 1.5 M in their HAeBe samples (Thé et al., 1994; Vieira et al., 2003),
despite such low-mass objects remaining largely convective for most of their pre-main
sequence lifetime.
Two useful definitions to introduce are the ‘birth line’ and the zero-age main
sequence (ZAMS). The birth line is the locus on the H-R diagram at which premain sequence stars first become visible. Upon reaching the birth line, the stars
have completed most of their accretion and cleared enough of their neighborhood
for their circumstellar material to become optically thin. The ZAMS is the locus
on the H-R diagram at which a star first arrives at the main sequence. At this
point the star has completed its contraction phase and begins core hydrogen burning.
Theoretical pre-main sequence evolutionary tracks, as well as the birth line and the
ZAMS are illustrated in Figure 1.4. Pre-main sequence lifetimes of HAeBe stars (the
time necessary to evolve from the birth line to the ZAMS) range from 107 years for
∼ 2 M stars to around 105 years for ∼ 6 M stars (Palla & Stahler, 1993).
Following the pre-main sequence evolutionary model calculations from the classic
paper of Palla & Stahler (1993), it is expected that stars with masses below 2.4
M cross the birth line fully convective and on the Hayashi evolutionary track5 .
Stars in the mass range 2.4 M to 3.9 M cross the birth line with a radiative
5
The Hayashi track is line along which a protostar begins its life, and collapses with nearly
constant effective temperature. Stars on the Hayashi track are fully convective, and depart from the
Hayashi track when radiative energy transport becomes important.
25
CHAPTER 1. INTRODUCTION
3
6.0 Msun
birt
h lin
5.0 Msun
e
log L (Lsun )
4.0 Msun
2
3.5 Msun
3.0 Msun
2.5 Msun
Hayashi track
2.0 Msun
1
2.5 Msun
ZA
S
M
0
4.2
4
Log T (K)
3.8
3.6
Figure 1.4: Pre-main sequence evolutionary tracks (solid lines) from Palla & Stahler
(1993) along with the birth line (dashed) and the zero age main sequence (ZAMS)
line (dotted) for a 10−5 M yr−1 initial accretion rate. Evolutionary tracks have
been labeled by mass, in solar masses. The Hayashi track portion of visible pre-main
sequence evolution is indicated. A typical pre-main sequence star becomes optically
visible at the birth line, then evolves towards the ZAMS.
CHAPTER 1. INTRODUCTION
26
core and convective outer regions, and become visible after the end of the Hayashi
track evolution. Stars with masses above 3.9 M begin their pre-main sequence life
fully radiative and thermally relaxed, well away from the Hayashi track. All stars
then evolve towards the main sequence by undergoing a thermal readjustment, if not
already thermally relaxed, and then contracting to the main sequence. The results
presented here assume a 10−5 M yr−1 initial accretion rate. In the case of larger
initial accretion rates, the birth line essentially moves up, allowing for longer premain sequence life times, and allowing more massive stars to exist in the observable
pre-main sequence phase (Palla & Stahler, 1993). The general evolutionary behavior
found by Palla & Stahler (1993) is consistent with that found by other authors (e.g.
Bernasconi, 1996).
There remains significant opportunity for further study of the complex physics
of HAeBe stars. The current work focuses on aspects of their magnetic fields and
connections to Ap/Bp stars. The role of magnetic fields in the formation of intermediate mass stars is barely understood. It is possible that magnetic fields play a role
in accretion in intermediate mass stars (Muzerolle et al., 2004), but this has yet to be
investigated in detail. Additionally, the role of magnetic fields in angular momentum
loss for intermediate mass stars is unknown. In the case of the slow rotating Ap/Bp
stars, ‘magnetic braking’ (transfer of angular momentum from the rotating star to the
surrounding material via the magnetic field) seems likely, however direct observational
confirmation of this has yet to be seen and the time-scale for the process is somewhat uncertain (Stȩpień & Landstreet, 2002). It remains unclear whether magnetic
HAeBe stars evolve into Ap/Bp stars, although circumstantial evidence supporting
this is mounting (Wade et al., 2007). If there is an evolutionary connection, and some
CHAPTER 1. INTRODUCTION
27
HAeBe stars are to become chemically peculiar, the timescale for the generation of
chemical peculiarities in HAeBe stars is unknown.
1.5
HD 72106
HD 72106 is a visual double star system with a 0.800 separation between the components (Hartkopf et al., 1996). The brighter component (HD 72106A) is often referred
to as the primary, and the dimmer component (HD 72106B) is often referred to as the
secondary. The system is located at a right ascension of 08h:29m:34.90s and declination of -38◦ :360 :2100 , placing it in the constellation Vela. The system has a combined
spectral class of A0IV and was identified as having an infrared excess, based on IRAS
(the Infrared Astronomical Satellite) data, by Oudmaijer et al. (1992). Blommaert
et al. (1993) provided additional infrared photometric measurements of the combined
system, but erroneously suggested that the object was a post-AGB (post-asymptotic
giant branch) star. Torres et al. (1995) observed the system photometrically in visible
light, as well as with low resolution optical spectroscopy, and noted the presence of
emission in the Hα Balmer line. They hypothesized that the system was associated
with the Gum Nebula.
The HD 72106 system was observed by the Hipparcos satellite (ESA, 1997).
A separation between the components of 0.80500 was found, at a position angle of
199.4◦ (relative to celestial north). The system was solved as a double star system,
giving a good quality solution. A parallax of 3.47 ± 1.43 mas (milli-arc seconds)
was found, with a large proper motion in right ascension of −5.81 ± 1.08 mas yr−1
(milli-arc seconds per year) and in declination of 8.73 ± 1.36 mas yr−1 . A Hipparcos
magnitude Hp = 8.937 ± 0.005 was observed for the primary and 9.734 ± 0.010 for
CHAPTER 1. INTRODUCTION
28
the secondary. No evidence for relative motion between the two components was
observed.
Fabricius & Makarov (2000) re-reduced the Hipparcos dataset, together with data
from the Tycho mission (a secondary mission on the Hipparcos satellite, ESA, 1997),
providing Tycho VT and BT magnitudes for both components of HD 72106. For the
primary they found a BT magnitude of 8.91 ± 0.01 and a VT magnitude of 9.00 ± 0.01
(BT − VT = −0.09 ± 0.014). For the secondary they found BT = 9.82 ± 0.02 and
VT = 9.62 ± 0.02 (BT − VT = 0.20 ± 0.03).
Vieira et al. (2003) observed the individual components and identified HD 72106B
as a HAeBe star. The authors associated it with the Gum Nebula star forming region.
They observed Hα emission, but noted that it was fairly weak for a HAeBe star. In
examining HD 72106B’s spectral energy distribution Vieira et al. (2003) found a very
low contribution from dust. This, they hypothesized, was due to HD 72106B being
an evolved HAeBe star that has cleared most of its circumstellar envelope.
Schütz et al. (2005) performed infrared spectroscopy of HD 72106B. They examined circumstellar abundances of a number of different molecules. Schütz et al.
(2005) found that the circumstellar material around HD 72106B was dominated by
the crystalline silicates forsterite and enstatite. They found that large grains of amorphous silicates (grain size ∼ 2.0 µm) and SiO2 were present, but that smaller grained
amorphous silicates (grain size ∼ 0.1 µm) were almost completely absent. When
comparing the IR spectrum of HD 72106B with a linear combination of spectra from
comets Hale-Bopp and Halley, Schütz et al. (2005) found an “unusually” good fit.
The authors comment that such a large fraction of processed dust is seen in only a
few Herbig Be stars. These results seem to support the hypothesis of Vieira et al.
CHAPTER 1. INTRODUCTION
29
(2003): that HD 72106B is a relatively evolved HAeBe star and has cleared most of
the dust in its vicinity.
Recently, Drouin (2005) and Wade et al. (2005) reported a detection of a magnetic
field in HD 72106A based on spectropolarimetry from FORS1 (FOcal Reducer/low
dispersion Spectrograph) at the Very Large Telescope and from ESPaDOnS (Echelle
SpectroPolarimetric Device for the Observation of Stars) at the Canada France Hawaii
Telescope. This was one of the earliest detections of a magnetic field in a possible
HAeBe star. A longitudinal field of +195±40 G (i.e. 4.9σ) was detected in the FORS1
spectrum. Remarkably, no significant magnetic field was detected in the secondary
(+65 ± 55 G was observed). In the ESPaDOnS spectrum no significant longitudinal
field was detected, but clear circular polarization was detected with a very high degree
of significance, unambiguously indicating the presence of a magnetic field.
A more accurate and robust analysis of the same FORS1 data by Wade et al.
(2007) did not result in a longitudinal magnetic field detection at 3σ confidence in HD
72106A. Wade et al. (2007) report longitudinal fields of 166 ± 70 G from Balmer lines
and −11 ± 91 G from metallic lines. They conclude that one cannot be confident of a
magnetic field detection at this level, however they note that ESPaDOnS observations
have shown a clear circular polarization signal, and that HD 72106A is definitely
magnetic.
Wade et al. (2005) report a mass for the primary of 2.4 ± 0.4 M and 1.75 ± 0.25
M for the secondary, based on the stars’ H-R diagram positions. They report an
age of approximately 10 Myr for the system. Wade et al. (2007) accept these values,
and deduce Teff = 11000 ± 1000 K and 3.5 ≤ log g≤ 4.5 for the primary, and Teff =
8000 ± 500 K and 4.0 ≤ log g≤ 4.5 for the secondary. Wade et al. (2005) also report
CHAPTER 1. INTRODUCTION
30
strong variability in absorption lines of the primary, and suggest that this may be due
to surface chemical abundance spots. In this thesis we find that the dramatic line
variability reported by Wade et al. (2005), and illustrated in their Figure 2, is largely
an instrumental artifact, due to the absence of atmospheric dispersion correction
in one of their two observations, as described in Section 2.3. However, in other
more recently acquired spectra we do see significant variability of line profiles, and
confidently interpret this as due to surface abundance inhomogeneities, as described
in Section 4.4. Wade et al. (2005) suggest that the observed variability is due to
rotational modulation, with a period near 2 days. Our results qualitatively support
this, with periodic variability due to stellar rotation providing a good phasing of
observations separated by years, but we derive a somewhat shorter rotation period,
as discussed in sections 3.6 and 4.2.
Due to its young age, probable binarity, and magnetic and chemical properties,
the HD 72106 system was considered to be a compelling target for further study. The
discovery of magnetic fields in HAeBe stars provides a tantalizing link to Ap and Bp
stars. However, an in depth study of several magnetic HAeBe objects is necessary.
Such studies can provide critical evidence to support or refute proposals of an evolutionary link between some HAeBe and Ap/Bp stars. The magnetic field geometries
of HAeBe stars, their rotational properties, and their surface chemistry can provide
such evidence. Study of chemical abundances in HAeBe stars can help understand
any link between HAeBe and Ap/Bp stars, as well as determine the time scale on
which abundance peculiarities develop. The details of the chemical peculiarities seen
could have important consequences for the diffusion model of abundance anomaly
formation.
CHAPTER 1. INTRODUCTION
31
The HD 72106 system is a prime target for such study. The components of HD
72106 have rather similar fundamental parameters. They have similar masses and
temperatures, and most likely identical ages. These similarities make the observed
differences all the more striking. The hints at a magnetic field in the primary, conclusively confirmed here with further observations from ESPaDOnS, place it in an
interesting class of objects. Both stars are very young, making the link to HAeBe objects, but they have spectra with very little emission, making them appealing objects
for spectroscopic analysis. The binarity of the system can provide further evolutionary constraints on the formation of magnetic fields and chemical peculiarities, given
the apparently normal secondary. Thus HD 72106 is a particularly interesting system,
situated at an important point in its evolution, and which is especially well suited to
detailed study.
This thesis presents such a detailed study of HD 72106, investigating photospheric
chemical abundances, rotational characteristics, magnetic properties, and surface
abundance distributions. The dynamical connection between the stars and their
evolutionary state is also discussed in detail. Chapter 2 discusses the collection of
observations of HD 72106 with ESPaDOnS, as well as the reduction of those observations. Chapter 3 discusses the analysis techniques used, particularly spectrum
synthesis, Least Squares Deconvolution, rotation period analysis, and Doppler Imaging. This chapter also describes the determination of fundamental parameters for
both stars, discusses the evolutionary state of the system, and analyses the evidence
for HD 72106 being a true binary system. Chapter 4 presents the major results of
this thesis: detailed mean surface abundances for both stars, the rotation period of
the primary, the magnetic field strength and geometry of the primary, and surface
CHAPTER 1. INTRODUCTION
32
abundance maps for the primary. Chapter 5 summarizes and discusses these results,
and draws conclusions, with implications for the formation of Ap/Bp stars.
Chapter 2
Observations
2.1
Observing Stellar Magnetic Fields
The observation of magnetic fields in stars can be challenging, but in many cases is
essential for understanding the physical processes at work in a star. The presence,
strength, and geometry of magnetic fields in stellar atmospheres are measured through
the atomic line splitting and polarization properties of the Zeeman effect.
2.1.1
Polarization
The polarization of light is commonly described in terms of the four Stokes parameters: I, Q, U, and V. Stokes I refers to the total amount of light in all polarization
states. Stokes Q is the difference between orthogonal linear polarization states. Stokes
U is the same as Stokes Q but with the basis set rotated by 45◦ . Stokes V is the difference between right and left circular polarization (Shurcliff & Ballard, 1964). This
is schematically illustrated in Figure 2.1.
33
34
CHAPTER 2. OBSERVATIONS
I =
+
Q=
-
U=
-
V=
-
or
+
or
+
Figure 2.1: Schematic diagram of the combinations of polarization states used to
calculate the Stokes parameters. I is the sum of polarizations, Q and U are the
differences between orthogonal linear polarizations at two different angles, V is the
difference between circular polarizations.
s The Stokes parameters can be described in a more rigorous fashion. Consider a
general electromagnetic wave E(t) as a superposition of two orthogonal (Cartesian)
components, perpendicular to the direction of travel, with magnitudes E1 and E2 :

 

i(φ1 +ωt)

E1 (t) A1 e
(2.1)
E(t) = 
,
=
i(φ2 +ωt)
A2 e
E2 (t)
where components 1 and 2 have phases φ1 and φ2 , and amplitudes A1 and A2 respectively. The frequency of the wave is ω and t is time. The Stokes parameters can then
be written as (Rees, 1987):
I = E1∗ E1 + E2∗ E2 = |E1 |2 + |E2 |2 = A21 + A22 ,
(2.2)
Q = E1∗ E1 − E2∗ E2 = |E1 |2 − |E2 |2 = A21 − A22 ,
(2.3)
U = E1∗ E2 + E2∗ E1 = 2 Re(E1∗ E2 ) = 2A1 A2 cos(φ1 − φ2 ),
(2.4)
V = E1∗ E2 − E2∗ E1 = 2 Im(E1∗ E2 ) = 2A1 A2 sin(φ1 − φ2 ),
(2.5)
CHAPTER 2. OBSERVATIONS
35
where ∗ refers to the complex conjugate. Thus I is the total intensity, Q is a measure of
linear polarization along the coordinate system, U is a measure of linear polarization
rotated 45◦ to the coordinate system, and V is a measure of circular polarization.
2.1.2
Zeeman Splitting
In the presence of a magnetic field, each atomic energy level is split into multiple sublevels with different polarization properties. This effect was first observed by Pieter
Zeeman in 1897, and it now bears his name. In the Zeeman effect, the degeneracy
in electron spin and angular momentum for one atomic transition is lifted by the
presence of an external magnetic field. Thus one observed atomic line becomes a
set of closely spaced lines, with a spacing approximately proportional to the field
strength. Not only is the line split, but the light emitted (or absorbed) by each
component of the line must possess a particular polarization, which is a function of
the magnetic field geometry. The shift in wavelength of the Zeeman components is
proportional to the magnetic field strength. Thus an observer, an astronomer for
example, can measure magnetic field strength by observing the wavelength shift of
the polarized components in a split absorption or emission line.
Fundamentally, the state of an electron in an atom is described by the following set
of quantum numbers: n (the principal), l (azimuthal or angular momentum), and s
(spin). These have the associated quantum numbers (eigenvalue for the z component
of the angular momentum L and spin S eigenvectors), ml and ms . The quantum
number l can range from 0 to n − 1 and for an electron, s = 12 . ml runs from −l to l,
and ms is ± 21 . The total angular momentum quantum numbers j and mj can be very
useful when the spin and angular momentum (l and s) are strongly coupled. In this
36
CHAPTER 2. OBSERVATIONS
case of L-S coupling, the total angular momentum quantum number can be written
as: j = l ± 12 , that is j = 1/2 . . . n − 1/2, and mj = −j . . . j (Griffiths, 1995).
In the ‘normal’ Zeeman effect, spin is not considered and a triplet of lines is
produced. This is really a special case of the ‘anomalous’ Zeeman effect discussed
below. In this simple case, with the magnetic field along the z axis, a electron
changing energy levels makes a transition in ml of ∆ml = −1, 0, or +1 (the only
allowed transitions). Due to the interaction with the magnetic field, these angular
momentum transitions do not all have the same energy. Thus the line is split into
three evenly spaced components, centered around the ∆ml = 0 transition at the zerofield wavelength (Ostlie & Carroll, 1996). Figure 2.2 illustrates both the splitting of
atomic energy levels, and the corresponding observed absorption line splitting.
No Magnetic Field
Magnetic Field
ml = 2
l=2
ml = 1
ml = 0
m l = -1
m l = -2
l=1
ml = 1
ml = 0
m l = -1
Figure 2.2: Illustration of atomic level splitting by the ‘normal’ Zeeman effect. The
top half of the figure shows the splitting of energy levels with different ml . The
bottom portion of the figure illustrates splitting in the observed (Stokes I) spectrum.
37
CHAPTER 2. OBSERVATIONS
In the ‘anomalous’ Zeeman effect, a transition in spin (ms ) as well as in ml occurs,
resulting in the formation of more than three lines. In this case it is more appropriate
to consider the total angular momentum, since there exists strong spin-orbit coupling1 . Transitions occur with ∆mj = −1, 0, or +1, similar to the normal Zeeman
effect. The critical difference is that the spacing, in terms of energy, between the
split levels depends on the l and j quantum numbers. The energy change (∆E) in
the presence of a magnetic field (B) due to the Zeeman effect is (Griffiths, 1995)
∆E = µB B gJ mj ,
where µB is the Bohr magneton (µB =
e~
)
2m
(2.6)
and gJ is the Landé g-factor (often referred
to as just the Landé factor). The Landé factor is given by (Griffiths, 1995)
gJ = 1 +
j(j + 1) − l(l + 1) + 3/4
,
2j(j + 1)
(2.7)
although in practice it is frequently determined experimentally, since the assumption
of spin-orbit coupling is often only approximate. Due to different quantum numbers,
different initial energy levels can be split into sets of sub-levels with different spacings.
This allows for a wider range of transition energies (although ∆mj is still 0 or ±1),
and hence more observable components to the Zeeman split line.
The ∆mj = 0 components in a Zeeman split line are referred to as the π components. When the magnetic field is oriented along an observer’s line of sight (the
‘longitudinal’ direction), these components are not observed. In this geometry, the
∆mj = ±1 components, referred to as the σ components, are observed to emit or
absorb left or right circularly polarized light, depending on whether ∆mj is +1 or
1
We consider only the Zeeman effect in the linear regime here. The Paschen-Back and intermediate field regimes are neglected. While less general, this is a safe treatment for the field strengths
encountered in Ap/Bp stars (Wade et al., 2000a).
CHAPTER 2. OBSERVATIONS
38
-1. In this case, the observed splitting and polarization is known as the longitudinal
Zeeman effect. If the magnetic field is oriented such that it is perpendicular (i.e.
transverse) to the observer’s line of sight (such that it lies in the plane of the sky),
the light emitted or absorbed from π components is observed to be linearly polarized, with a polarization angle parallel to the magnetic field. In this geometry the σ
components are also observed to be linearly polarized, but with a polarization angle
perpendicular to the field direction. This case of the Zeeman effect is referred to as
the transverse Zeeman effect (e.g. Wade et al., 2000a). An observation of three Zeeman split lines in total intensity and circular polarization is presented in Figure 2.3
Because the transverse Zeeman effect is a secondary property, the linearly polarized
signal is much weaker than the circularly polarized signal (Wade et al., 2000a). Additionally, in the case of a rotationally broadened line profile, it is often much easier to
detect the circular polarization signature from the Zeeman effect than the wavelength
splitting in total intensity. Thus, in our investigations, we have focused on circular
polarization. Analysis of circular polarization is generally the method of choice for
studying magnetic fields in stellar astronomy.
2.2
Instrumentation
Observations for this thesis were collected at the Canada-France-Hawaii Telescope
(CFHT) with the Echelle SpectroPolarimetric Device for the Observation of Stars
(ESPaDOnS) instrument.
39
CHAPTER 2. OBSERVATIONS
1.2
V/I
1.1
Flux
I
1
0.9
0.8
777.8
778
778.2
778.4
Wavelength (nm)
778.6
Figure 2.3: An ESPaSDOnS spectrum illustrating line splitting and polarization in
HD 94660 due to the Zeeman effect. The lower line is a total intensity (Stokes I)
spectrum and the upper line is a circular polarization (Stokes V divided by Stokes
I) spectrum, shifted vertically for clarity. Since HD 94660 has a very low v sin i,
individual components of the split lines can be seen in the total intensity spectrum,
as well as the change in polarization across the line profiles in the circular polarization
spectrum.
CHAPTER 2. OBSERVATIONS
2.2.1
40
Canada France Hawaii Telescope
The CFHT is a 3.58 meter telescope, with a Prime Focus/Cassegrain combination
design, located atop Mauna Kea on the Big Island of Hawaii. The CFHT is situated at
an altitude of 4204 m, a latitude of +19◦ 490 41.8600 and a longitude of 155◦ 280 18.0000 .
The CFHT can act as a f/8 Cassegrain focus or a f/4 prime focus, and historically has
also had a Coudé mode of operation. For our observations, and all observations with
ESPaDOnS, the telescope operates in Cassegrain focus mode. Other instruments,
notably WIRCam and MegaPrime, require the telescope to operate in prime focus
mode. The telescope is controlled remotely from an operations room one floor below
the dome.
2.2.2
ESPaDOnS
The Echelle SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS)
instrument is a high resolution spectropolarimeter that consists of two major parts.
Mounted at the Cassegrain focus of the CFHT is a polarimeter module that analyses
the polarization of incoming light, and passes orthogonal polarization states of the
analysed light into a fiber-optic bundle. The optical fibers carry the light to a benchmounted echelle spectrograph located in the Coudé room of the CFHT, which is
located one floor below the operations room.
This design allows for the spectrograph to be well-insulated thermally and vibrationally, thereby improving the spectroscopic stability of the instrument. Placing the
spectrograph in the Coudé room also reduces the stress on the telescope. Mounting the polarimeter in the Cassegrain module on the telescope allows for immediate
CHAPTER 2. OBSERVATIONS
41
analysis of polarization, reducing spurious and instrumental effects caused by the optical train. The location of the Cassegrain module also allows for the easy inclusion
of dedicated calibration and guiding optics. The most significant drawback of this
two-piece design is that passing light into and along the optical fibers can result in
a loss of flux. Problems with overly reflective and damaged fiber optic bundles have
been encountered with the ESPaDOnS instrument. For example, between January
and June 2005 a damaged fiber bundle resulted in a loss of throughput efficiency of
about 1.0 mag. At present all known transmission problems have been corrected.
ESPaDOnS has a nominal resolving power R = λ/∆λ = 68000 when operating
as a spectropolarimeter, and 80000 when operating as a spectrograph. The instrument provides nearly continuous wavelength coverage from 3700 Å to 10500 Å in 40
echelle orders. The peak throughput of the system, as measured from the telescope’s
aperture, is approximately 15% to 20%. The entrance aperture of the ESPaDOnS
polarimeter is a 1.6 arcsecond (00 ) pinhole with a physical size of 0.22 mm.
Polarimeter
The Cassegrain-mounted polarimeter module houses an atmospheric dispersion corrector, a compact CCD for guiding, and a calibration wheel, as well as optics for
analysing the polarization of the incident light.
The Atmospheric Dispersion Corrector (ADC) consists of a pair of rotatable
prisms that are used to counteract the wavelength dependence of the refractive index
of air. Light passing through the earth’s atmosphere is dispersed by differential refraction, causing a wavelength-dependent distortion of astronomical images. The ADC
corrects for this distortion, ensuring that no extra blue or red light is systematically
CHAPTER 2. OBSERVATIONS
42
lost outside of the pinhole (see e.g. Mynne & Worswick, 1986).
The guiding CCD collects light that is reflected from a tilted mirror surrounding
the pinhole. Guiding images are collected in relatively short exposures (0.5 − 2 seconds), allowing for nearly real-time guiding corrections. By ensuring that the amount
of light escaping around the edge of the pinhole remains symmetric, the guiding computer or observer can ensure that the target star remains centered.
The calibration wheel is used to replace the stellar beam with light from calibration
sources. The standard calibration lamps are a pair of flat field tungsten lamps and a
thorium-argon arc lamp for wavelength calibration.
The polarization analyzing optics consist of two half-wave Fresnel rhombs, a
quarter-wave Fresnel rhomb, and a Wollaston prism, as illustrated in Figure 2.4.
The half-wave Fresnel rhombs are placed on either side of the quarter-wave Fresnel
rhomb, and are rotatable about the optical axis of the instrument. The half-wave
Fresnel rhombs, as the name suggests, provide a π/2 retardation to the phase of
the polarization, thus for example, swapping left and right circular polarization. The
quarter-wave Fresnel rhomb converts circular polarization to linear polarization, much
like a regular quarter-wave plate, by providing a π/4 phase retardation (Shurcliff &
Ballard, 1964). Fresnel rhombs were chosen for ESPaDOnS because they behave in
a less wavelength dependent fashion than quarter-wave plates. This is essential, as
it allows ESPaDOnS to perform nearly achromatic polarization analysis over a large
wavelength range (Alecian, 2007).
The half-wave rhombs, as mentioned, are rotatable about the optical axis of the
instrument, while the quarter-wave rhomb and the Wollaston prism are fixed. This
freedom of rotation allows for the optics to provide any desired retardation to the
CHAPTER 2. OBSERVATIONS
43
Figure 2.4: Schematic diagram of the polarization analysing optics in the Cassegrain
module of ESPaDOnS. The individual optical components are labeled. Light enters
from the Cassegrain focus, travels through the optical train, and exits into the fiber
optic bundle for transmission to the spectrograph.
incoming beam of light. Thus one can transform any initial orthogonal pair of polarization states into any other orthogonal pair, before the light passes into the Wollaston
prism, by selecting the appropriate angles for the half-wave rhombs. The choice of
angle effectively reduces to a system of two equations (say the retardation of the
horizontal and vertical linear polarization states) and two unknowns (the angles of
the two rotatable rhombs). This allows one to easily analyse light, so as to produce
Stokes I, Q, U or V spectra, without having to add or remove optical components
from the path of the beam.
Upon exiting the last Fresnel rhombs, the light enters the Wollaston prism. The
Wollaston prism splits linearly polarized light into its orthogonal states and produces
two beams (Shurcliff & Ballard, 1964). Thus for the analysis of circular polarization,
the light is first converted into linear polarization (π/4 retardation) with the Fresnel
CHAPTER 2. OBSERVATIONS
44
rhombs, then the Wollaston prism produces one (linearly polarized) beam containing
light that was originally left circularly polarized, and another (linearly polarized)
beam that was originally right circularly polarized. These beams are then injected
into separate optical fibers for transmission to the spectrograph.
The optical fiber bundle consists of three fibers, only two of which are used at
a time. The bundle is approximately 30 m long, with fibers that have a 0.1 mm
core diameter and 0.11 mm diameter including cladding. ESPaDOnS uses low-OH
H-treated fibers produced by CeramOptec, allowing for very good throughput over a
wide spectral domain (Donati, 2007). Peak throughput of the fibers is around 90%.
Spectrograph
Light entering the spectrograph from the optical fibers first encounters a BowenWalraven image slicer (Donati, 2007). The image slicer transforms the round images
from the two fiber heads into a set of three narrow elongated slices (when the instrument is operating in spectroscopic ‘star only’ mode this is doubled to six slices). This
is done by slightly redirecting portions of the beams from the fiber heads with a set
of two prisms and a parallel plate.
The light is then collimated with a parabolic collimator and passed to the diffraction grating. The diffraction grating is an R2 echelle grating with a 204x408 mm
ruled area, 79 lines/mm, and a 63.4 degree blaze angle. The dispersed light then
passes through a prism cross-disperser to separate the spectral orders. Light finally
enters the f/2 camera and is collected by a 2000x4500 pixel CCD detector with 13.5
µm square pixels (Donati, 2007). The optical path through the major components of
the spectrograph is illustrated in Figure 2.5
CHAPTER 2. OBSERVATIONS
45
This design of the spectrograph provides a total of 40 orders projected onto the
CCD. Wavelength coverage runs from 372 nm to 1029 nm with three small gaps
at 922.4 - 923.4 nm, 960.8 - 963.6 nm, and 1002.6 - 1007.4 nm. The CCD pixels
correspond to 2.6 km s−1 and are extracted at 1.8 km s−1 (that is 0.6923 CCD pixels
per spectral pixel, see Section 2.4 for the extraction process) (Donati, 2007). The
spectrograph has a peak throughput of about 40% to 45%, giving the total peak
throughput of the telescope and instrument of 15% to 20%.
Figure 2.5: Schematic diagram of the major components in the ESPaDOnS spectrograph. The individual optical components are labeled. Light enters from the fiber
optic bundle, passes through the spectrograph, and is collected by the CCD in the
camera.
2.3
Observations
Observations were collected during four different observing missions (runs) between
February 2005 and March 2007. All runs used the same instrument, ESPaDOnS,
and all observations were collected and reduced in the same manner. Approximately
half the observations (those in the March 2007 run) were obtained during a time
CHAPTER 2. OBSERVATIONS
46
allocation for which the author of this thesis wrote the proposal and was the principal
investigator. A copy of this proposal is included in Appendix A. The journal of
observations is reported in Table 2.1.
The initial setup at the start of a night of observations is heavily automated.
The appropriate directory structure for acquired images and data reduction must be
created. The dewar of liquid nitrogen used to cool the CCD must be filled. After that
is complete the automatic focusing routine is executed and the telescope is ready to
acquire calibration images.
A full set of calibration frames was collected at the beginning and end of each
observing night. This consisted of 3 bias frames, one thorium argon lamp exposure,
20-40 internal tungsten lamp flat field frames, and about 4 Fabry-Perot exposures.
A sample flat field frame and a sample thorium argon frame are presented in Figure
2.6. The bias frames are used to correct for detector background and readout noise.
However, rather than averaging over multiple bias frames (to reduce noise in the bias
frame), the ESPaDOnS bias frames are averaged over 8 pixel by 8 pixel squares, before subtraction from science images. The thorium argon arc lamp exposure provides
a set of well known emission lines used for wavelength calibration. The wavelength
calibration is refined in science images with the use of telluric lines (terrestrial atmospheric absorption lines due to water vapor) in the science images themselves. The
flat field frames are also used to identify the orders on the CCD chip, and fit them
to produce a geometric model of the frame. The Fabry-Perot frames, which contain
a large number of very sharp ‘lines’, are used to fit the shape and orientation of the
‘slit’ image on the CCD.
All observations were made in polarimetric mode, as ‘Stokes V’ observations, using
47
CHAPTER 2. OBSERVATIONS
UT Date
21 Feb.
22 Feb.
09 Jan.
11 Jan.
12 Jan.
12 Jan.
11 Feb.
11 Feb.
12 Feb.
13 Feb.
13 Feb.
13 Feb.
02 Mar.
02 Mar.
02 Mar.
03 Mar.
04 Mar.
05 Mar.
05 Mar.
05 Mar.
09 Mar.
09 Mar.
05
05
06
06
06
06
06
06
06
06
06
06
07
07
07
07
07
07
07
07
07
07
HJD
Component Integration
(-2 450 000) (HD 72106)
Time (s)
3422.9730
3423.9248
3745.02967
3747.02034
3747.99629
3748.01496
3777.87860
3777.95149
3778.86172
3779.87202
3779.95052
3779.98127
4161.77282
4161.80256
4161.90282
4162.85713
4163.83561
4164.84650
4164.88387
4164.90961
4168.85791
4168.90947
A&B
A&B
A&B
A&B
A
B
A&B
A&B
A&B
A&B
A&B
A&B
A&B
A&B
A&B
A&B
A&B
B
A
A&B
A&B
A&B
2400
2400
2400
3200
1200
1200
2000
2400
2400
2400
2400
2400
2400
2400
2400
2400
2400
3200
2400
1600
2400
2400
Peak S/N Peak S/N
I
V
222
201
219
143
149
76
238
253
282
184
107
128
254
265
297
208
322
209
248
277
283
272
208
186
219
116
149
88
221
239
259
175
15
79
248
255
285
195
308
200
241
272
278
258
Table 2.1: Log of observations of the HD 72106 system obtained with ESPaDOnS.
Column 1 is the date the observation was obtained on, column 2 is the heliocentric
Julian date of the observation, and column 3 indicates which component(s) of the
double star were observed. Integration times are given in column 4 and refer to the
complete set of 4 sub-exposures required to produce each Stokes V spectrum. Peak
signal-to-noise ratios are given for the Stokes I spectra in column 5 and Stokes V
spectra in column 6, and are reported for 1.8 km s−1 spectral pixels.
CHAPTER 2. OBSERVATIONS
48
Figure 2.6: Sample flat field (left) and thorium argon (right) calibration frames.
Both frames have a zoomed in detail in their lower right. The flat field frame
illustrates the geometry of the spectral orders, with the closely spaced pairs corresponding to opposite polarization states. The thorium argon frame illustrates
the ‘slit’ shape from individual emission lines. The heavily saturated lines are
not used for data reduction. Images: N. Manset, Canada-France-Hawaii Telescope,
http://www.cfht.hawaii.edu/Instruments/Spectroscopy/Espadons/
CHAPTER 2. OBSERVATIONS
49
the 40 second ‘normal’ readout mode. Each ‘Stokes V’ observation produces both a
real circular polarization Stokes V spectrum and a total intensity Stokes I spectrum.
A complete Stokes V observation consists of 4 sub-exposures, between each of which
the angles of the half-wave rhombs are chosen so as to produce either a π/4 or a
−π/4 retardation, in the sequence: π/4, −π/4, −π/4, π/4. Exposure times for each
observation (i.e. the complete set of 4 sub-exposures) are given in Table 2.1.
All observations listed in Table 2.1 (with one exception, discussed below) were
made with the atmospheric dispersion corrector in place and the fiber agitator on.
The fiber agitator jiggles the fiber optic bundle to eliminate any modal noise in the
cable. Modal noise is caused by the interference of different optical modes in a fiber.
Varying the physical geometry of the fiber changes the preferred modes and can
average out this noise.
As indicated in Table 2.1, both components of the HD 72106 system were usually
observed together, due to the small separation of the components. For the observations of the combined system, the photo-center of the system was centered in the
pinhole. In these observations atmospheric conditions (worse than 0.500 seeing) were
insufficient to resolve the individual components properly. However, with a 1.600 pinhole and a separation between components of 0.800 , keeping both components in the
pinhole did not pose a problem.
Observations of the individual components were made on nights with particularly
good seeing. With seeing better than 0.400 it is possible to resolve the two components
of the binary. With careful guiding, one can exclude one component from the pinhole.
Manual guiding and careful monitoring are necessary to ensure flux from the undesired
component does not enter the pinhole.
CHAPTER 2. OBSERVATIONS
50
Ultimately, two of the observations were rejected as unsuitable for science. The
observation at HJD 2453779.95052 (second observation on 13 Feb. 2006) was rejected
due to cloud cover during part of the exposure, which resulted in a very low signal-tonoise ratio (S/N). The observation at HJD 2453422.9730 (21 Feb. 2005) was rejected
because it was the one case in which the ADC (atmospheric dispersion corrector) was
accidentally not used. The observation was of the combined HD 72106 A & B system.
As a result of the atmospheric distortions, the amount of flux recorded from each of
the two components varied with wavelength. Thus some observed spectral lines were
primarily contributed by the primary and some were primarily from the secondary.
This effect cannot be reliably corrected for, and thus the observation was discarded.
The observation collected without the use of the ADC was discussed by Wade et
al. (2005). The authors erroneously diagnosed very strong variability in HD 72106
due to this observation. While we do find substantial variability in HD 72106A, the
differences in spectra seen by Wade et al. (2005) are primarily a result of the unused
ADC, not intrinsic variability in either star.
2.4
Reductions
Basic data reductions were performed with the Libre-ESpRIT data reduction package.
This is a nearly-automatic dedicated data reduction package for ESPaDOnS, which
performs a complete calibration and optimal spectrum extraction. Continuum normalization of the resulting unnormalized spectra was performed with a ‘home made’
routine, tailored by hand to HD 72106A and B. This routine, described in Section
2.4.1 and included in Appendix B, fits low-order polynomials to the continuum in each
spectral order and then divides the spectrum by the polynomial, hence normalizing
CHAPTER 2. OBSERVATIONS
51
the continuum to unity. Observations of the primary were reconstructed by subtracting the (assumed constant) spectrum of the secondary, using appropriate weighting,
from the spectra of the combined system, as described in Section 2.4.2.
2.4.1
Spectrum Extraction and Normalization
The Libre-ESpRIT data reduction package is based on the ESpRIT (Echelle Spectra
Reduction: an Interactive Tool) package (Donati et al., 1997), but it has been tailored
specifically for ESPaDOnS and has been further automated. This package allows for
fully reduced spectra to be available within minutes after an exposure sequence is
complete.
Libre-ESpRIT begins by performing a bias subtraction. The software then calculates the geometry of the orders on the CCD, producing a best fit model, and
pixel-to-pixel sensitivity variations are corrected with the mean flat field. Flat field
and thorium argon frames are illustrated in Figure 2.6. The slit geometry is then
modeled with the Fabry-Perot images and bad pixels are rejected. A wavelength
calibration is applied using the thorium argon frame, defining the dispersion relation
and providing a relationship between CCD position and spectral wavelength. The
robustness of the calibration is verified using regions of wavelength with overlap of
different spectral orders.
Once the initial calibration is complete, optimal spectrum extraction is performed,
producing a 1D spectrum from each of the four 2D sub-exposures. The spectra from
the individual sub-exposures are combined as follows. To produce the (total intensity)
Stokes I spectrum (I), the flux from individual sub-exposures is added:
I = i1,⊥ + i1,k + i2,⊥ + i2,k + i3,⊥ + i3,k + i4,⊥ + i4,k ,
(2.8)
52
CHAPTER 2. OBSERVATIONS
where ik,⊥ and ik,k are the two spectra of orthogonal polarization obtained in subexposure k. The polarization rate V /I is given by
R−1
V
=
,
I
R+1
(2.9)
where R = RV and
RV4 =
i1,⊥ /i1,k i4,⊥ /i4,k
,
i2,⊥ /i2,k i3,⊥ /i3,k
(2.10)
as described by Donati et al. (1997). This method requires more sub-exposures than
is strictly necessary to produce V /I, however it corrects for systematics much better
than is otherwise possible. In particular, swapping optical paths in two sets of subexposures allows for the suppression, at least to first order, of spurious polarization
signals due to variations in optical path, including CCD sensitivity. Repeating this
process allows for the canceling out of time dependent effects, again to first order (see
Semel et al., 1993; Donati et al., 1997).
The ‘null polarization’ spectrum N/I is also calculated by replacing R in equation
4
2.9 with RN
(Donati et al., 1997) where:
4
RN
=
i1,⊥ /i1,k i2,⊥ /i2,k
.
i4,⊥ /i4,k i3,⊥ /i3,k
(2.11)
In this case, rather than multiplying together coherent polarization states as for V /I,
we divide coherent polarization states, hence canceling out any real signal. N/I is
useful for ensuring that a signal in V /I is in fact real. In general, N/I should contain
no signal. If a signal is observed in the N/I spectrum then some systematic error is
likely present, and it must be corrected if the data is to be used.
The individual ik,⊥ and ik,k spectra are extracted by fitting a (model) spectrum
to the observed CCD orders. In this ‘inverse problem’ the model spectrum is processed through the previously calculated model slit and order geometries, and then
CHAPTER 2. OBSERVATIONS
53
compared, via χ2 , to the observed CCD image. The autocorrelation matrix from this
fitting process is used to derive error bars for each individual pixel. Error bars on the
final I and V /I spectra are then computed from the individual sub-exposure error
bars. Once the optimal extraction process is complete, a wavelength correction to
the heliocentric rest frame is applied. A final optional step is continuum normalization. The quality of the Libre-ESpRIT normalization is not sufficiently good for the
detailed analysis that we will perform, therefore we only use unnormalized output.
Continuum Normalization
In general, it is very difficult to accurately define the continuum level in absolute
units in high resolution stellar spectroscopy. The continuum recorded on the CCD
is a product of the true continuum of the star, the interstellar transmission, the terrestrial atmospheric transmission, the system transmission of both the telescope and
instrument, and the response of the CCD. Furthermore, most of the important information for stellar spectroscopy is contained in individual absorption (or emission)
lines. Thus, rather than deal with the difficult problem of absolutely fixing the continuum, it is common practice to normalize the continuum to unity. Then the properties
of absorption and emission lines are analysed relative to the normalized continuum.
The general procedure is to fit an appropriate (but ad hoc) function through regions identified as continuum, then to divide the entire observed spectrum by the
function. A smooth (low-degree) polynomial function is often used, since the continuum changes very slowly with wavelength compared to the (relatively sharp) absorption lines of interest. In echelle spectroscopy, relatively steep order profiles make
normalization somewhat more challenging, particularly at the ends of orders.
54
CHAPTER 2. OBSERVATIONS
A home made routine written in Fortran and tailored for HD 72106A & B was
used. A copy of the normalization routine can be found in Appendix B, Section B.1.
This routine follows the basic strategy outlined above, with low-degree polynomials
as the fitting function. The function used (y(x)) for a degree of n is
y(x) =
n
X
a i xi ,
(2.12)
i=0
with a0 . . . an as the free parameters to be fit. The routine operates on one spectral
order at a time. This avoids the problem of order overlap, and the much higher-degree
polynomials needed to deal with changes in the continuum level between orders. It
also has the advantage that the robustness of the normalization for each order can be
checked in the overlapping regions of successive orders.
Points in the continuum are identified by first taking a running average (over 17
pixels) of the spectrum, smoothing out the noise. Then the pixel with the greatest
flux in a moderately large (200 pixel) region is identified. The size of the region is
chosen to be larger than the spacing between most lines; thus there usually is some
real continuum to be identified in each region. These points are then used to find the
polynomial fit. The points chosen are much fewer than the actual number of points
in the continuum, however they are chosen with some care so as to ensure that they
are representative of the true continuum.
The degree of the polynomial used is determined manually for each order of the
spectrum. Usually a degree between 3 and 5 was chosen. Additionally, regions of the
spectrum with emission lines or particularly broad absorption features can be excluded
from the fit by hand. Figure 2.7 illustrates fits to the continuum in a segment of
unnormalized ESPaDOnS spectrum, as well as the subsequently normalized spectrum.
Note that the S/N in this segment is 220 (based on spectral pixels) and the error bar
55
CHAPTER 2. OBSERVATIONS
1.05
1
1
Normalized Flux
Flux
0.95
0.9
0.95
0.9
0.85
0.85
0.8
500
505
510
Wavelength (nm)
515
520
500
505
510
Wavelength (nm)
515
520
Figure 2.7: An unnormalized spectrum with polynomial fits to the continuum (smooth
lines) is presented in the left frame. Two spectral orders are shown, using different
colors. In the right frame, the same spectrum, now with the normalization applied,
is shown. A typical error bar for this segment of spectrum is illustrated in the right
frame.
56
CHAPTER 2. OBSERVATIONS
associated with each pixel is approximately 0.0045, as illustrated in Figure 2.7.
2.4.2
Spectrum Reconstruction
The majority of our spectra correspond to the combined light of both components of
the HD 72106 system. Analysis of the system requires that we recover the spectra of
the individual components.
The recovery of the spectrum of HD 72106A from the combined spectrum relies on
our observation that the spectrum of the secondary remains stable over long periods
of time2 . This is illustrated by the two nearly identical spectra of HD 72106B collected
in January 2006 and March 2006. In these two spectra, all of the Balmer lines other
than Hα were identical within the uncertainties. In Hα we see clear emission which
displays some variability. The metallic lines are also all identical except for the O
line at 7771 Å, in which a small amount of variable emission is seen. The stability
between observations is illustrated in Figure 2.8. This stability is not surprising; later
results show the secondary to be a normal Herbig Ae star (see Section 4.1.2), and
thus one would not expect significant variability outside of emission lines.
Having established the stability of the spectrum of the secondary, we can subtract
it from each of the combined spectra, weighted by its relative contribution, to obtain
spectra of the primary. We begin by modeling the intensity IT at any point in the
combined spectrum according to
IT LT = I 1 L1 + I 2 L2 ,
(2.13)
where I2 refers to the normalized observed flux of the secondary and L2 refers to the
2
There are two exceptions to this: the Hα Balmer line and the O 7771 Å line, both of which are
observed to be variable in emission.
57
CHAPTER 2. OBSERVATIONS
luminosity of the secondary; thus the product I2 L2 gives the total observed flux of
the star as a function of wavelength. I1 refers to the normalized flux of the primary
(our unknown), while L1 is the luminosity of the primary. IT and LT are the total
normalized flux of the system and the total luminosity of the system respectively.
The total luminosity can be written as LT = L1 + L2 . It is elementary to rearrange
equation 2.13 to obtain I1 :
I1 = I T
L2
1+
L1
− I2
L2
.
L1
(2.14)
The magnitude difference between components is well known from Hipparcos data
(ESA, 1997; Fabricius & Makarov, 2000). In the Hipparcos V band the apparent
magnitude of the primary is 8.937 and the secondary is 9.734 (ESA, 1997). In the
Tycho V band the apparent magnitudes are 9.00 and 9.62 respectively (Fabricius &
Makarov, 2000). This allows us to easily determine the ratio L2 /L1 using the standard
relation:
L2 /L1 = 100(m1 −m2 )/5 ,
(2.15)
where m1 − m2 is the difference in apparent magnitude between the primary and
secondary. A bolometric correction is not necessary here, as we are interested in
the relative luminosities in a particular wavelength range - the spectral range of
the ESPaDOnS instrument. The procedure described here, when used with V band
photometry, is not expected to work outside of the bandpass of the V filter.
This procedure allows us to reconstruct 18 spectra of the primary from the combined spectra, rather than using just the 2 observations of the primary on its own.
However, we cannot generally use the same technique to reconstruct spectra of the
secondary. The primary displays clear variability of its absorption lines. Hence the
observations of the (variable) primary, obtained at an arbitrary date, cannot be used
CHAPTER 2. OBSERVATIONS
58
with observations of the combined system to reconstruct spectra of the secondary.
In addition, subtracting a reconstructed spectrum of the primary from its original
combined spectrum simply recovers the original observation of the secondary used in
the initial reconstruction. However, regions of the primary’s spectrum that exhibit
no detectable variability can be used to recover small parts of the spectrum of the
secondary. This was used to further verify the non-variability of the secondary.
2.4.3
Quality Control
Because of the complexity of the procedure described in Section 2.4.2, we have performed a number of checks to ensure the accuracy of the reconstructed spectra.
The first check performed was a comparison of the observed spectra of just the
primary (obtained on nights of excellent seeing, Julian date 2453747.99629 and Julian
date 2454164.88387) with each other, and those of just the secondary (Julian date
2453748.01496 and Julian date 2454164.84650) with each other. Careful monitoring
at the time of observation makes us confident that the observations of one component
were not contaminated by light from the other. This was confirmed by comparing
observations of the same object. At the level of the S/N (76 in the worst observation),
the two observations of the secondary were identical outside of Hα and OI 7771 Å,
the two clear emission lines observed in the star. A careful examination of every
angstrom of these two spectra by eye revealed no other variations above the noise
level. Figure 2.8 illustrates the stability of the spectra of HD 72106B between our
observations, obtained 417 days apart. Thus we conclude that the spectrum of the
secondary is constant within the noise, outside of emission lines.
Regions in which the spectra of the primary displayed little to no variability
59
CHAPTER 2. OBSERVATIONS
1
Flux
0.8
0.6
0.4
480
485
Wavelength (nm)
490
1
0.95
Flux
0.9
0.85
0.8
0.75
0.7
450
451
452
453
Wavelength (nm)
454
455
Figure 2.8: Sample Hβ and metallic lines in observations HD 72106B on its own. The
black spectra were obtained on 12 Jan. 2006, the red (gray) spectra were obtained
on 5 Mar. 2007. A very good correspondence can be seen between the observations.
60
CHAPTER 2. OBSERVATIONS
1
Flux
0.8
0.6
0.4
480
490
485
Wavelength (nm)
1
0.95
Flux
0.9
0.85
0.8
0.75
0.7
451
452
453
Wavelength (nm)
454
455
Figure 2.9: Sample Hβ and non-variable metallic lines in observations HD 72106A on
its own. The black spectra were obtained on 12 Jan. 2006, the red (gray) spectra were
obtained on 5 Mar. 2007. In these ‘stable’ regions of spectrum, a good correspondence
can be seen.
61
CHAPTER 2. OBSERVATIONS
were used to reconstruct spectra of the secondary, as discussed in Section 2.4.2. In
general, reconstructed spectra of the secondary are not useful due to unaccounted
for variability in the primary’s spectrum. However, in these segments of spectral
stability, the process can reliably be performed. In these small regions, no variability
was seen in any reconstructed observation of the secondary, down to the noise level
of the observation. The stability of one such region is illustrated in Figure 2.10. The
lack of variability in these regions provides further support for the conclusion that
the secondary is non-variable, at least to the noise level in our observations.
Flux
1
0.9
0.8
450
451
452
453
Wavelength (nm)
454
455
Figure 2.10: Sample reconstructed spectra of the secondary, in a stable region of
the primary’s spectrum. The black line is based on the observation from 4 March,
2007 and the red line is based on the second observation from 11 February, 2006. In
general, reconstructed spectra of the secondary are not useful due to unaccounted for
variability in the primary’s spectrum. However, in a few spectral regions where the
primary displays little to no variability, one can reliably reconstruct spectra of the
secondary. In these regions we seen no variability in the secondary, providing further
support to the conclusion that HD 72106B is non-variable.
CHAPTER 2. OBSERVATIONS
62
A similar examination of spectra from the primary found that they displayed identical Balmer line profiles, and a number of identical metal lines. This is illustrated in
Figure 2.9, which compares stable segments of spectra in observations of HD 72106A
on its own. However, unlike the spectrum of the secondary, changes in a significant
number of metal lines were observed. These changes are attributed to intrinsic variability of the primary’s spectrum. Several examples of this are illustrated in Figure
2.11 using spectra of just HD 72106A. Asymmetric line profiles were observed in the
majority of these cases (particularly clear in the segment of spectrum centered at
∼443 nm, shown in Figure 2.11), leading us to infer intrinsic variability. The variability being restricted to only lines of some species (such Si) also suggests that it
is intrinsic, rather than an artefact of the observation, reduction, or reconstruction
techniques.
Reconstructed spectra of the primary were compared to observations of the primary on its own, as well as to each other. An observation of HD 72106A compared
to an observation of the combined system and a reconstructed spectrum are shown
in Figure 2.12. Neither the core depth nor the wing shape of the Balmer lines seen
in the combined spectra match the observations of the primary on its own. Reconstructed spectra, on the other hand, provide excellent matches to the spectra of the
primary on its own. Additionally, the majority of metallic lines in the combined
spectra do not match the spectra of the primary on its own. However, reconstructed
spectra (at the correct rotation phase of the primary3 ) provide nearly perfect matches.
3
The rotation phases of our observations and the rotation period of HD 72106A are discussed in
detail in Sections 3.6 and 4.2. In these sections we find that the variability observed here is due to
rotational modulation of HD 72106A. As the star rotates the visible portion of the stellar surface
changes, and hence the observed properties vary. The rotational phase φ is defined φ = t/P , where
t is the time of observation and P is the rotation period. Thus φ runs from 0.0 to 1.0 during one
rotation.
63
CHAPTER 2. OBSERVATIONS
1
0.95
Flux
Cr II
0.9
Fe II
Fe II
Fe II
Fe II
Fe II
Fe II
Fe II
0.85
Si II
0.8
Si II
503
504
505
Wavelength (nm)
506
507
1.02
1
0.98
0.98
0.96
0.96
0.94
Ti II
0.92
Mg II
Fe II
0.92
443
443.5
Wavelength (nm)
Fe II
Cr II
0.9
Mg II
442.5
Si I
0.94
Mg II
0.9
0.88
Flux
Flux
1
444
623
623.5
Fe II
Cr II
624
624.5
Wavelength (nm)
625
Figure 2.11: Variable metallic lines in observations of HD 72106A on its own. The
black spectra were obtained on 12 Jan. 2006, the lighter (red) spectra were obtained
on 5 Mar. 2007. Clear variability can be seen in lines of some species, while other lines
have not varied substantially between observations. Some lines show clear variations
in strength, some in radial velocity, and some vary in both. Major contributors to
the stronger lines are labeled, in order of importance.
CHAPTER 2. OBSERVATIONS
64
Reconstructed spectra at widely separated rotation phases do not display matching
metallic line profiles, due to the profile variability established above, but the Balmer
lines do not vary. Pairs of reconstructed spectra at nearly the same phase (and at
diametrically opposite phases) are shown in Figure 2.13. From this we conclude that
the spectrum recovery process is necessary to produce usable spectra, and that the
process provides very good results.
Comparing recovered observations of HD 72106A at the same phase or opposite
phases, as determined in Section 4.2, highlights the accuracy of the reconstructed
spectra, as well as the intrinsic variability in HD 72106A. Figure 2.13 presents the
same spectral region, in one case with two observations separated by 0.50 in phase
and in another case with observations separated by only 0.03. The spectra in both
comparisons were obtained over a year apart. In the first case, strong variability can
be seen, as well as asymmetric line profiles. In the second case, while asymmetric
line profiles are seen, the spectra are virtually identical. This provides further clear
evidence for the variability of HD 72106A. As well, this further demonstrates that the
spectrum reconstruction technique is sufficiently accurate to provide nearly identical
spectra of the primary at the the same rotation phase. Thus systematic errors due
to spectrum reconstruction are small, at or below the noise level.
65
CHAPTER 2. OBSERVATIONS
Flux
0.8
0.6
0.4
486
487
488
Wavelength (nm)
489
1
Flux
0.95
0.9
0.85
0.8
525
526
Wavelength (nm)
527
528
Figure 2.12: Comparison of Hβ and metallic lines in an observation of just HD
72106A, a combined observation of HD 72106A & B, and a reconstructed spectrum
of HD 72106A. The black spectrum is of just the primary, from 5 March, 2007. The
darker blue (gray) spectrum is an observation of the primary and secondary combined,
obtained on 11 January, 2006. The green (lighter gray) spectrum is the reconstructed
spectrum of the primary from the combined spectrum. The two observations were
obtained at nearly the same rotation phase of the star, with only a difference in phase
of 0.04 cycles. The reconstructed spectrum matches the observed primary spectrum
almost perfectly, despite the original combined spectrum it was produced from being
a poor match.
66
CHAPTER 2. OBSERVATIONS
1
Flux
0.95
0.9
0.85
501
502
Wavelength (nm)
503
504
501
502
Wavelength (nm)
503
504
1
Flux
0.95
0.9
0.85
Figure 2.13: Reconstructed spectra of HD 72106A at different rotational phases. In
the top frame, spectra with a separation in phase of 0.50, from 9 March, 2007 (black)
and 12 February, 2006 (gray/red) are compared. Strong variability and asymmetric
line profiles can be seen. In the bottom frame, spectra with a separation in phase of
0.03, from 9 March, 2007 (black) and 11 February, 2006 (gray/green) are compared.
While asymmetric line profiles are still present, the spectra are almost identical.
Chapter 3
Analysis
Having demonstrated our ability to accurately recover the individual spectra of the
two components of HD 72106, we now discuss the techniques used to analyse the
spectra and interpret the properties of the stars. These techniques will allow us to
determine the fundamental parameters of both stars, the surface chemical abundances
of both stars, the rotational properties of the primary, the magnetic field geometry
and strength of the primary, and the surface chemical abundance distribution for the
primary.
3.1
3.1.1
Basic Physical Parameters
Effective Temperature and Surface Gravity
Determining fundamental physical parameters for HAeBe stars is very challenging.
Initially, we require effective temperature (Teff ) and surface gravity (log g in logarithmic units) to characterize the stellar atmosphere. Photometric methods, using
67
CHAPTER 3. ANALYSIS
68
photometry in several different bands and an appropriate photometric calibration, are
unreliable. This is due to contributions to the energy distribution from circumstellar
emission and absorption from dust and gas (e.g. van den Ancker et al., 1998). Fitting
a model spectral energy distribution to an observed one can in principle be used to
derive a temperature, however no appropriate observations exist for the individual
components of HD 72106. Balmer line fitting is the remaining commonly used technique. This is also an unreliable technique, as many HAeBe stars display emission
in their Balmer lines, and Balmer lines generally don’t provide a unique solution for
Teff and log g. However, the HD 72106 system is fairly evolved, to the point that the
primary is at about the zero age main sequence, and hence one expects little emission
in the Balmer lines of either component. Observationally, the primary displays no
observable emission in Balmer lines; the secondary has observable emission only in
Hα and the OI 7771 Å lines in our spectra.
Balmer line fitting was performed by Wade et al. (2005) for both the secondary
and the primary of HD 72106. Following Wade et al. (2005), we attempted to constrain Teff and log g using the Balmer lines in the observed spectra. The observed
Balmer lines were compared to theoretical models computed with the ATLAS9 code
(Kurucz, 1993), calculated with solar metallicity model atmospheres. The theoretical
spectra, were convolved with a Gaussian instrumental profile, with a full width at
half-maximum adjusted to match the resolution of the observations. Balmer lines in
ESPaDOnS spectra are generally recorded in multiple spectral orders, as mentioned
earlier. Normalization of echelle spectra between orders can be unreliable. The normalization is further complicated by the lack of true continuum for large portions of
the orders due to the broad Balmer line wings. Consequently, the modeling results
CHAPTER 3. ANALYSIS
69
from the ESPaDOnS Balmer lines were viewed as tentative at best.
Observations from the FORS1 (FOcal Reducer/low dispersion Spectrograph) instrument at the Very Large Telescope (VLT) in Chile were obtained from Wade et
al. (2005). The observations were obtained with the grism GRIS 600B+12, providing
a resolving power of about 815 from 345 nm to 590 nm. Individual spectra of both
HD 72106A and HD 72106B were obtained. As FORS1 is a single order spectrograph,
these spectra were collected in a single order. Therefore, they do not suffer from the
normalization effects that afflict the ESPaDOnS Balmer lines. Balmer line fitting
was performed on these spectra by Wade et al. (2005) and Drouin (2005). For the
primary they find: Teff = 11000 ± 1000 K and 3.5 ≤ log g ≤ 4.5, (best fit Teff =
11000 K, log g = 4.0). For the secondary they find Teff = 8000 ± 500 K and 4.0 ≤
log g ≤ 4.5, (best fit Teff = 8000 K, log g = 4.5). We repeated the fitting procedure
with the FORS1 spectra ourselves, using the method outlined above, and arrived at
results identical to Wade et al. (2005) and Drouin (2005) for the primary. For the
secondary we find the temperature range 7500 K to 9000 K more realistic, with a
best fit value of 8000 K at log g= 4.5. The log g of the secondary we find to be
in the range 4.0 to 4.5 range, with smaller values being more appropriate for hotter
temperatures. These values are also consistent with the values determined by Balmer
line fitting in the ESPaDOnS spectra. Sample fits from of the FORS1 Balmer lines
are shown in Figure 3.1 for the best fit values, as well as the Teff = 8750 K, log g =
4.0 model discussed below.
When spectrum synthesis was performed for HD 72106B, it became apparent that
the Balmer line best fit Teff and log g were not able to reproduce the detailed line
profiles. The spectrum synthesis procedure is discussed in more detail in Section
70
CHAPTER 3. ANALYSIS
1.1
A
B
1
Normalised flux
0.9
0.8
0.7
0.6
0.5
0.4
0.3
4260
4280
4300 4320 4340 4360 4380
Wavelength (angstroms)
4400 4260
4280
4300 4320 4340 4360 4380
Wavelength (angstroms)
4400
Figure 3.1: Sample Balmer line fits of Hγ for HD 72106A in frame A and HD 72106B
in frame B. The solid lines represent best fit models and the points represent the
observed spectra. In frame A the model is Teff = 11000 K log g = 4.0. In frame B
the darker (red) line represents a Teff = 8000 K log g = 4.5 model and the lighter
(green) line represents a Teff = 8750 K log g = 4.0 model.
3.3.2, with results discussed in Section 4.1. After extensive spectrum modeling in 7
independent spectral windows, with the range of Teff and log g allowed by the error
bars from Balmer line fitting, it was found that the values Teff = 8750 K and log g
= 4.0 gave a substantially better fit to lines across the spectrum of HD 72106B. Even
with chemical abundance, projected rotational velocity, and microturbulence as free
parameters, an 8000 K temperature resulted in systematic inconsistencies between
the model and observed spectra, as did a log g of 4.5. As a consequence, we adopt
Teff = 8750 K and log g = 4.0 as the most accurate effective temperature and surface
gravity for the secondary. Best fit synthetic spectra for both the initial Teff = 8000
K, log g = 4.5 model and the best fit Teff = 8750 K, log g = 4.0 model are shown
in Figure 3.2. However, a temperature of 9000 K and log g of 4.0 provides almost
as good a fit as the 8750 K model. Including the full range of Teff and log g which
71
CHAPTER 3. ANALYSIS
provide at least marginally acceptable fits to the modeled metallic lines in HD 72106B,
we find a conservative uncertainty in Teff of 500 K and in log g of 0.5. While the best
fit values of Teff = 8750 ± 500 K and log g = 4.0 ± 0.5 produce a somewhat poorer
fit to the Balmer lines, the fit to metallic lines is improved substantially, hence we
conclude that the higher temperature and lower log g are more accurate.
1
Flux
0.95
Fe I
Cr I
Fe I
Ti II
Ti I
Cr I
Ti II
Cr I
Fe II
Cr I
0.9
Fe II
Fe I
Fe II
Fe II
Ti I
0.85
Ti II
Fe II
0.8
4520
4530
Wavelength (angtsroms)
4540
Figure 3.2: Sample best fit spectra for HD 72106B at both the original Teff = 8000
K and log g = 4.5 in (in red/darker gray) as well as the best fit Teff = 8750 K
log g = 4.0 (in green/lighter gray). Major contributors to each line are labeled, in
order of importance. This represents a segment of a larger 100 Å region that was fit
simultaneously. For both Teff and log g values, v sin i, microturbulence, and chemical
abundances were fit. Even with these free parameters, clear discrepancies between
lines of different ionization states of the same element can be seen in the Teff =
8000 K and log g = 4.5 case. The Teff = 8750 K log g = 4.0 model resolved these
discrepancies, and hence we adopt it as the best fit effective temperature and surface
gravity.
72
CHAPTER 3. ANALYSIS
3.1.2
Mass, Radius and Age
With an effective temperature and luminosity it is straightforward to place a star on
the Hertzsprung-Russell (H-R) diagram, which allows for the determination of other
physical parameters (by comparison with evolutionary models) such as mass and age.
In the case of the HD 72106 system, Hipparcos parallax measurements exist, allowing
one to determine luminosity using the inverse square law.
The observed Hipparcos parallax of the HD 72106 system is 3.47 ± 1.43 mas (ESA
1997), which indicates that the system is situated at a heliocentric distance of 288+202
−84
pc.
Using this distance, one can convert the apparent (observed) magnitude to an
absolute magnitude, with the standard distance modulus relation:
Mv = mv − 5 log d + 5,
(3.1)
where d is the heliocentric distance of the object, mv is the apparent magnitude and
Mv is the absolute magnitude (Ostlie & Carroll, 1996). To obtain the luminosity of an
object one requires a bolometric (i.e. integrated across all wavelengths) magnitude.
The bolometric absolute magnitude (Mbol ) is calculated from the absolute magnitude
by addition of the appropriate bolometric correction (BC) (Ostlie & Carroll, 1996):
Mbol = Mv + BC.
(3.2)
Before applying a bolometric correction we first converted the Tycho V magnitude,
available for HD 72106 from Fabricius & Makarov (2000), into a standard Johnson V
magnitude. This is necessary because bolometric correction tabulations are general
for the Johnson V band. The Tycho V and Johnson V filters are very similar, so a
linear conversion can be used unless extreme accuracy is required. In the case of the
73
CHAPTER 3. ANALYSIS
HD 72106 system the uncertainties in distance dominate any uncertainties due to the
conversion from Tycho to Johnson V magnitudes. We therefore used the empirical
relation:
V = VT + 0.09(BT − VT ),
(3.3)
where V is the Johnson V magnitude, VT is the Tycho V magnitude and BT is the
Tycho B magnitude (ESA, 1997).
For the primary, the bolometric correction relation of Landstreet et al. (2007) was
used. This calibration is tailored specifically for magnetic chemically peculiar A and
B type (Ap and Bp) stars, hence it is the most appropriate calibration available for
the primary1 . Due to the strong chemical over-abundances seen in Ap/Bp stars, many
abnormally strong metallic lines are present in their spectra. These lines absorb flux
in one wave band, which is then re-emitted in other wavebands. This ‘line blanketing’
effect results in Ap/Bp stars having spectral energy distributions that differ somewhat
from normal A and B stars, hence they require a special bolometric correction. The
bolometric correction BCAp (Teff ) (in magnitudes) from Landstreet et al. (2007) is
given by:
BCAp (Teff ) = −4.891 + 15.147θ − 11.517θ 2 ,
(3.4)
where θ = 5040.0/Teff .
For the secondary, the bolometric correction for main sequence stars from Gray
(2005) was used. The correction is as follows:
BC = − 64741.46 + 67717 log Teff − 26566.141 log2 Teff
3
(3.5)
4
+ 4632.926 log Teff − 303.0307 log Teff ,
1
The strong chemical peculiarities of the primary, and their similarity to those of Ap/Bp stars,
are discussed further in Section 4.1
74
CHAPTER 3. ANALYSIS
where Teff is the effective temperature of the star.
With the bolometric magnitudes for both stars one can finally determine the
luminosity (L) of the two stars, using the relation:
L
= 10(Mbol −Mbol )/2.5 ,
L
(3.6)
where L is the luminosity of the sun and Mbol
is the bolometric magnitude of the
sun (L = 3.826×1026 W, Mbol
= 4.76 mag) (Ostlie & Carroll, 1996). The luminosity
+31
of the primary determined from equation 3.6 is 23 −13
L , and the luminosity of the
+12.7
secondary is 9.9 −5.6
L .
The radius of each star can be computed directly from the Stefan-Boltzmann
equation:
2
L = 4πR2 σTeff
,
(3.7)
where R is the stellar radius and σ is the Stefan-Boltzmann constant. This produces
a radius of 1.3 ± 0.6 R for the primary and 1.4 ± 0.6 R for the secondary.
The stars can be placed on an H-R diagram using the derived values of Teff and L,
as shown in Figure 3.3. The H-R diagram positions can be compared with theoretical
evolutionary tracks and isochrones, allowing the determination of mass and age. We
use the pre-main sequence evolutionary model calculations of Palla & Stahler (1993).
We find the age of the HD 72106 system to be between 3 and 13 Myr, based on the
position of the secondary in the H-R diagram. The mass of the primary is 2.4±0.4 M
and mass of the secondary is 1.9 ± 0.2 M . The position of the primary falls mostly
below the zero age main sequence line (ZAMS), but is still consistent with the ZAMS
within 1σ uncertainties. If one compares the H-R diagram position of the primary to
main sequence evolutionary tracks, one can constrain its age to be less than 8.2 log
yr (∼160 Myr). However, if the system is truly a binary, the constraint on the age
75
CHAPTER 3. ANALYSIS
of the system derived from the secondary (3 to 13 Myr) is much more precise. Error
bars associated with Teff , log g, luminosity, radius, and mass are all approximately
1σ.
3
Birthli
ne
4M
2.5
log L
2
sun
3M
sun
ZAM
S
1.5
2M
1 Myr
sun
3 Myr
1
0.5
4.1
5 Myr
10 Myr
1.5 M
HD 72106A
HD 72106B
sun
4
Log Teff
3.9
3.8
Figure 3.3: An H-R diagram containing both components of the HD 72106 system.
Isochrones (solid lines) and evolutionary tracks (dashed lines), as well as the birth
line (for an accretion rate of 10−5 M yr−1 ) and the zero age main sequence (ZAMS)
line, are taken from Palla & Stahler (1993).
It is worth noting that, while the absolute luminosities of the components are
poorly determined, their ratio is very well determined. This is because the major
uncertainty in the absolute luminosity is the distance to the stars, and the stars are
situated at the same distance. Thus the uncertainties on the radii or masses derived
for the two stars are not independent. If the primary has a higher mass then, due to
76
CHAPTER 3. ANALYSIS
the highly accurate ratio of luminosities, so does the secondary. The spacing between
the components in log L on the H-R diagram must remain essentially fixed.
Given the importance of the evolutionary status of these stars, a somewhat more
in-depth discussion is necessary. The presence of circumstellar material around the
HAeBe secondary assures us of its very young age, justifying the use of pre-main
sequence evolutionary tracks. For the primary we do not have the same strong justification. Given its H-R diagram position, the primary could be as young as ∼ 3
Myr (as measured from the birth line), and in that case have ∼
1
4
of its pre-main
sequence lifetime remaining. The most likely case (from ‘best fit’ positions) is that
the system is ∼ 10 Myr old, thus the primary has just entered the main sequence
and the secondary is on its final approach to the ZAMS. In this case the primary
would have spent ∼ 6 Myr on the main sequence, giving it a fractional age on the
main sequence (τ ) of 0.01. In the oldest limiting case, the system is ∼ 13 Myr old
and secondary is just reaching the ZAMS. In this scenario, the primary has been on
the main sequence for up to ∼ 9 Myr and has fractional age of 0.015. Further study,
particularly a more accurate distance measurement, is necessary to more precisely
determine the evolutionary status of the primary. Regardless, if HD 72106A is not
on the pre-main sequence, it is certainly one of the youngest main sequence stars of
its type.
3.2
Binarity
Determining whether the HD 72106 system is truly a binary or just a double star
(i.e. an accidental conjunction of 2 stars at different distances along the line of sight)
is critical. If the system is a binary it allows us to constrain the age of the primary
CHAPTER 3. ANALYSIS
77
much more precisely than would otherwise be possible, as demonstrated in Section
3.1.2. Additionally, it implies that both components formed from approximately the
same material, making HD 72106 an interesting system from the point of view of
stellar magnetic and chemical evolution.
As mentioned previously, the system has a (projected) separation of 0.800 , and
thus is fairly widely separated. Given the Hipparcos parallax of 3.47 ± 1.43 mas, this
implies a minimum possible physical separation of 232 ± 96 AU.
The Hipparcos solution for the system finds that the stars have the same parallax
and proper motions. In the spectrum fitting (to be described in Section 3.3) we
included heliocentric radial velocities for each star as a free parameter. From this
fitting, we find identical radial velocities of 22 ± 1 km s−1 for both components. Thus
the stars are in the same point in space and moving together (in three dimensions),
strongly suggesting that they are in fact physically associated.
For the system to be a true binary, the stars must be orbiting one another. We
must check if the apparent lack of relative motion is consistent with the stars being
gravitationally bound. Assuming a circular orbit, and that the minimum possible
separation is the true separation (r), one can easily calculate the relative velocity of
the components of the binary (v). This is given by (Ostlie & Carroll, 1996):
2 1
2
v = G(m1 + m2 )
,
−
r a
(3.8)
where m1 and m2 are the masses of the primary and secondary respectively, G is
the gravitational constant, and a is the semimajor axis of the orbit of the reduced
mass about the center of mass of the system; in our circular case a = r. This
equation follows directly from the virial theorem. The true separation and orbits of the
components are unknown, and cannot be determined without accurate measurements
CHAPTER 3. ANALYSIS
78
of the relative motion of the components. In this assumed geometry, the relative
velocity of the stars is 4.0 ± 0.9 km s−1 and the orbital period is 1700 ± 800 years.
If one considers the case in which system is inclined with the orbital plane along
the line of sight and, as assumed above, the observed separation of the stars is the full
physical separation in a circular orbit, then one can calculate the maximum radial
velocity possible. In this geometry, illustrated in Figure 3.4, the relative velocity
between the stars is completely projected along the line of sight, and hence is larger
than the observed radial velocities. However, if the plane of the orbit is tilted by only
15◦ then the values become consistent with the observed velocities, within uncertainty.
Furthermore, if the stars’ physical separation is larger than we have assumed then
the velocities can be consistent with smaller inclination angles.
Figure 3.4: A schematic diagram of the orbital geometries considered for HD 72106.
For the first case discussed in Section 3.2, with the orbital plane aligned along the
line of sight, the observer would be looking along the dot dashed line. For the second
case discussed, the observer would be looking down on the page, towards the ‘X’.
In the other extreme, if the plane of orbit of the stars lies in the plane of the sky, as
CHAPTER 3. ANALYSIS
79
illustrated in Figure 3.4, then an observed relative proper motion of 2.9±1.4 mas yr−1
should be seen. One can calculate this simply by converting the relative motion given
above into a proper motion across the sky, at the appropriate distance. The result of
this calculation is marginally consistent with the Hipparcos proper motion reported
in Section 1.5. While the observed relative proper motion is zero, the uncertainties
on this value are significant. This results in a difference between the two values of
1.3σ. Again, if the physical separation of the stars is larger, or if the orbit is inclined
away from ‘face on’, the discrepancies become smaller.
Dr. Brian Mason at the United States Naval Observatory kindly provided a list of
separation and position angle observations from the Washington Double Star Catalog,
dating back to 1902 (private correspondence, Mason, 2006). These have been compiled
from a number of sources in the literature, such as Hartkopf et al. (1996) and the
Hipparcos mission (ESA, 1997), as well as a large number of observatory bulletins that
are no longer readily accessible, such as the Republic Observatory at Johannesburg
Circular. These observations indicate no change in the separation of the components,
but hint at a possible change in position angle of roughly +10◦ . Unfortunately, the
accuracy of the older observations in this catalog is unclear. If a 10◦ position angle
change over 90 years (1902-1992) has occurred, that is very much consistent with
the range of possible relative motions between components of HD 72106. We shall
consider the scenario presented above, in which the orbital plane is in the plane of
the sky and the stars are minimally separated on circular orbits. In this case the
change in position angle per year can be trivially calculated from the relative motion
and separation of the two stars. This produces a change of 19 ± 9◦ in 90 years. This
is in good agreement with the ∼ 10◦ value. Again, the theoretical value would be
CHAPTER 3. ANALYSIS
80
somewhat smaller if the orbit were inclined further or if the physical separation were
larger. There is clearly a large range of models that could fit this observation.
Generally one ought to consider elliptical orbits, however with the limited observations available for HD 72106 it is difficult to do so with any accuracy. Considering an
orbit with an eccentricity 0.5 and the system at periastron, no major changes are seen
to the above results, and a wide range of models are still allowed. The inclinations
discussed above are restricted only by roughly another 5◦ .
One final constraint is that the stars must have consistent positions on the H-R
diagram. As discussed in Section 3.1.2 and shown in Figure 3.3, this is the case.
The ages of the stars are consistent, implying that the system can be co-eval, hence
providing further evidence for binarity.
Thus we conclude that HD 72106 is almost certainly a true binary system. The
observations are, admittedly, not sufficient to guarantee that the stars are physically
associated with complete certainty. However, there is very strong evidence that this
is truly a binary system. The stars are at the same position in space, moving in the
same (three dimensional) direction at the same speed, and allow for a wide range of
gravitationally bound orbits. While we cannot determine the geometry of the orbit,
the small relative radial velocity and possible change in position angle suggest that
the binary may be closer to ‘face on’ than to ‘edge on’. The physical parameters of HD
72106A and HD 72106B, as well as some of the orbital constraints, are summarized
in Table 3.1
81
CHAPTER 3. ANALYSIS
HD 72106A
Teff (K)
log g
R (R )
M (M )
Age
HD 72106B
11000 ± 1000
8750 ± 500
4.0 ± 1.0
4.0 ± 0.5
1.3 ± 0.6
1.4 ± 0.6
2.4 ± 0.4
1.9 ± 0.2
10+3
−7 Myr
Orbital Constraints
Minimum possible separation
Maximum Relative velocity
Minimum Orbital period
232 ± 96 AU
4.0 ± 0.9 km s−1
1700 ± 800 years
Table 3.1: Fundamental physical parameters for HD 72106A and B, and orbital constraints on the system. The orbital constraints derived assume that the minimum
possible separation between the stars is their true physical separation, as well as
assuming circular orbits for both stars.
3.3
Spectrum Synthesis and Fitting
The observed total intensity (Stokes I) spectrum of a star contains a tremendous
amount of information about the photosphere of the star. In recent years the most
common, and arguably the most accurate, method of accessing information about
conditions near the surface of a star has been to model the spectrum in detail. Computer modeling of spectra can provide accurate information about surface chemical
abundances, projected rotation velocity (v sin i), velocity fields and the magnetic field.
Effects of chemical stratification near the surface of the star are sometimes detectable
(e.g. Babel, 1992; Ryabchikova et al., 2002; Kochukhov et al., 2006). Magnetic field
strength, effective temperature and surface gravity also play important roles in the
formation of absorption lines, but are best constrained before extensive metallic line
modeling is undertaken, so as to simplify the parameter space of possible models.
CHAPTER 3. ANALYSIS
82
In this work, modeling was performed using the ZEEMAN2 spectrum synthesis code (Landstreet, 1988; Wade et al., 2001). Fitting of the model spectrum to
the observed spectrum was performed using a Levenberg-Marquardt χ2 minimization
method.
3.3.1
ZEEMAN2
ZEEMAN2 is a local thermodynamic equilibrium (LTE) spectrum synthesis code that
solves the fully polarized radiative transfer equations. The code is descended from
the ZEEMAN code, written by Landstreet (1988) and then updated by Wade et al.
(2001). ZEEMAN2 is written in FORTRAN 77, and calculates Stokes I, Q, U, and
V spectra for an arbitrary set of rotational phases of a rotating magnetic star.
The ZEEMAN2 code takes as input a pre-computed plane parallel model atmosphere, as well as a table of atomic line data for the spectral line profiles to be
computed. The model atmospheres used in this thesis were computed using Atlas9
(Kurucz, 1993) for solar abundances and 2 km s−1 microturbulence. Atomic data
was extracted from the Vienna Atomic Line Database (VALD) (Kupka et al., 1999).
The “extract stellar” utility in VALD was used with a line depth threshold of 0.01.
For the primary, a Teff of 11000 K was used with a log g of 4.0; a 2 km s−1 microturbulence was used to simulate the desaturation effect of the magnetic field2 , and
very strong chemical over-abundances were included, to ensure the completeness of
the line list. For the secondary a Teff = 8500 K and log g= 4.0 were used with a 2
km s−1 microturbulence. Initially strong chemical over-abundances were included to
2
Magnetic desaturation is the increase in equivalent width seen in stronger absorption lines due
to the presence of a magnetic field. Such saturated lines, which are no longer on the linear part of
their curve of growth, are split by the Zeeman effect. This increases the wavelength range over which
the line can absorb light, and hence the total amount of light absorbed, if the line is saturated.
CHAPTER 3. ANALYSIS
83
ensure completeness of the line list, however when the abundances in the secondary
proved to be normal, line requests were altered to only use solar abundances for line
selections.
The ZEEMAN2 code calculates Zeeman splitting patterns of atomic lines using
the total angular momentum (j) quantum number for the lower and upper levels
of the transition. Experimental Landé factors for both levels are used to calculate
the strength of the splitting. Partition functions are calculated from the polynomial
approximation of Bolton (1970, 1971) for many lighter elements up to Zr. For heavier
elements, the approximations of Aller & Everett (1972) (for many neutral and singly
ionized rare earths), Cowley & Adelman (1983) (for a variety of heavier species),
and Cowley (1994) (for doubly ionized rare earths) are used. Species not included in
these sources use the approximations of Irwin (1981). Ionic populations are calculated
with the Saha equation for ions up to the triply ionized state. Radiation damping,
quadratic Stark damping and van der Waals damping are included, using constants
supplied by VALD. If the constants are not available, various approximations are used
(see Wade et al., 2001). Line opacity and anomalous dispersion profiles for metallic
elements are calculated using Voigt and Faraday-Voigt functions, implemented with
the numerical recipe of Humlicek (1982).
Continuous opacities are calculated from bound-free and free-free transitions for
neutral H, neutral He, and H− , as well as Rayleigh scattering from neutral H, and
electron scattering. These opacity sources are implemented using a variety of standard
algorithms, as detailed by Wade et al. (2001).
Numerical integration of the polarized radiative transfer equations is performed
using the semi-analytic method of Martin & Wickramasinghe (1979). The algorithm
CHAPTER 3. ANALYSIS
84
described by Martin & Wickramasinghe (1979) has been augmented to better handle
a number of special cases by Landstreet (1988) and Wade et al. (2001).
The magnetic field is included as an axisymmetric oblique multipole, with terms
up to octupole. Surface chemical abundance inhomogeneities can be described in a
simple parameterized fashion, usually by rings in magnetic latitude. Vertical chemical
stratification can be described using a simple two-step model. These last two features
were not used in this project, as there was no clear evidence for vertical chemical
stratification, and surface inhomogeneities were investigated in a more detailed fashion
with Doppler Imaging (see Section 3.7).
At each rotational phase, the visible disk of the star is divided into several hundred
to several thousand surface elements of approximately equal projected area. The
radiative transfer equation is integrated through the model atmosphere, with the local
magnetic field vector and chemical abundance included, for each of these elements
in their frame of reference. Large scale velocity fields, particularly v sin i, can then
easily be applied by Doppler shifting the local spectrum computed for individual
surface elements. The final step in the calculation of the synthetic flux spectrum is
to convolve the disk-integrated spectrum with an instrumental profile (assumed to be
Gaussian, with a width appropriate to the observed spectra).
The ZEEMAN2 code as been widely tested over a number of years and is now
considered very robust (e.g. Wade et al., 2001; Bagnulo et al., 2003; Folsom et al.,
2007; Silvester, 2007).
CHAPTER 3. ANALYSIS
3.3.2
85
Fitting Procedure
For both components of the HD 72106 system, surface chemical abundances and
v sin i, as well as microturbulence for the secondary, were determined by fitting a
synthetic spectrum from ZEEMAN2 to an observed spectrum. The high S/N observation of the primary at JD 2453747.99629 and the secondary at JD 2454164.84650
were used. Effective temperature and surface gravity were determined as described
in Section 3.1.1. The magnetic field of the primary was determined as described in
Section 3.5 and assumed to be null for the secondary, since there is no evidence for
the presence of a magnetic field in HD 72106B (see Section 4.3). As previously mentioned, no attempt was made to model the surface abundance inhomogeneities of the
primary. Therefore, the chemical abundances of the primary derived here are, strictly
speaking, only mean surface abundances for one phase. However, a comparison of the
JD 2453747.99629 spectrum to the average of all our spectra of HD 72106A shows a
particularly good correspondence. Fitting synthetic spectra to both the average observed spectrum and the JD 2453747.99629 spectrum gives a difference in the inferred
abundances between the two spectra of at most 0.05 dex. Thus the abundances presented here are roughly representative of the global mean surface abundances of HD
72106A. A detailed investigation of surface abundance inhomogeneities is discussed
in Section 3.7, with results described in Section 4.4.
Initially, for each star, direct comparison of the synthetic and observed spectra by
eye was used to determine approximate best fit values for relevant abundances, v sin i
and microturbulence. Once approximate values were determined, either from manual
fitting or fitting of another spectral region of the same object, the values were passed
as initial values to an automatic Levenberg-Marquardt χ2 minimization routine.
CHAPTER 3. ANALYSIS
86
The Levenberg-Marquardt fitting routine uses the recipe of Press et al. (1992).
The routine is designed as a Fortran 90 wrapper around the preexisting ZEEMAN2
code, thus leaving the ZEEMAN2 source code virtually unaltered, allowing for easy
upgrades as ZEEMAN2 is improved. Comparisons of the results of the LevenbergMarquardt fitting routine to careful fits by eye show very good agreement, well within
the uncertainties. Best fit model spectra, found using the Levenberg-Marquardt routine, are shown in Figures 3.5 and 3.6; further examples can be seen in Figures 4.1
and 4.3.
Despite the fact that χ2 values are calculated, they cannot be used to determine
reliable quantitative uncertainty estimates. The problem of spectrum synthesis is
sufficiently non-linear that something as simple as the covariance matrix cannot reliably be used. Moreover, the uncertainties are often dominated by uncertainties in
spectrum normalization or errors and omissions in the atomic line data, which are
not reflected in the observational error bars. This produces large values of χ2 , and
makes a formal uncertainty based on changes in χ2 unrealistic. Consequently, we base
uncertainties on the standard deviation of best fit values from a number of independently fit spectral regions. This accounts for intrinsic noise in the data, uncertainties
in atomic line data, and uncertainties in continuum normalization, as well as uncertainties (although not systematic errors) in other parameters such as Teff and log g.
In cases where the abundance of a particular element was determined using only a
few lines, uncertainties were based on a conservative estimate of the change in abundance necessary to shift the synthetic spectrum beyond any noise in the observed
spectrum, or any possible normalization errors. These uncertainty values represent
approximately a 2σ confidence level, and are indicated by an asterisk (*) in Table
87
CHAPTER 3. ANALYSIS
4.2. All other uncertainties in abundances, v sin i, and microturbulence are reported
in Table 4.2 at the 1σ confidence level.
Good, unique fits to the observed spectra are generally achievable. Figure 3.5
shows a good sample fit of a synthetic to an observed spectrum of the primary,
Figure 3.6 shows a good fit to an observation of the secondary. Figures 4.1 and 4.3
present additional high quality fits.
1
0.95
Flux
0.9
0.85
0.8
Fe
Ti
Cr
Fe
Cr
Fe
Cr
Fe
Cr
Fe
Fe
Cr Fe
Fe
Fe
Cr
4500
4510
Fe
4520
4530
Wavelength (angstroms)
4540
1
Flux
0.9
Cr Cr
Si
0.8
0.7
Fe
Cr
Ti
4550
Fe
Si
Cr
Fe
Cr
Cr
Fe
Cr
Fe
Fe
Cr
Fe
Cr
Cr
Cr
4560
4570
4580
Wavelength (angstroms)
4590
4600
Figure 3.5: Sample synthetic spectra fit to observations of HD 72106A, in two independently fit spectral regions. Major contributors to each line have been labeled, in
order of importance. The smooth solid line is the best fit spectrum in this region, the
dashed line is a spectrum computed with solar chemical abundances. These represent
sections of larger 100 Å regions that were fit simultaneously. The parameters for
these spectra were determined using the Levenberg-Marquardt fitting routine.
88
CHAPTER 3. ANALYSIS
1
Flux
0.9
0.8
Ti
Cr
Fe
Fe
Fe
Fe
Cr
Fe
Fe
Ti
Fe
Cr
Ba
Ti
Fe
Cr
Fe
Ti
0.7
4500
4510
4520
4530
4540
Wavelength (angstroms)
4550
4560
1
Flux
0.95
0.9
Ti
Fe
Y
Cr
Fe
Cr
Fe
Fe
Cr
Fe Fe
Cr
0.85
0.8
5180
Fe
Ti
Mg
5190
5200
5210
5220
Wavelength (angstroms)
5230
5240
Figure 3.6: Sample synthetic spectra fit to observations of HD 72106B, in two independently fit spectral regions.. Major contributors to each line have been labeled, in
order of importance. The smooth solid line is the best fit spectrum in this region, the
dashed line is a spectrum computed with solar chemical abundances. These represent
sections of a larger 100 Å regions that were fit simultaneously. The parameters for
these spectra were determined using the Levenberg-Marquardt fitting routine.
CHAPTER 3. ANALYSIS
89
In the case of HD 72106B, poor fits to the observed spectrum were found initially,
despite including v sin i, microturbulence, and chemical abundance as free parameters. Eventually it was found that the Teff and log g values from Balmer line fitting
were insufficiently constrained for accurate spectrum synthesis. Seven independent
regions of spectrum were fit for all allowed Teff and log g values from Balmer line
fitting, in steps of 500 K in Teff and 0.5 in log g. An extra step in temperature at
8750 K was included when both 8500 K and 9000 K (at log g= 4.0) were found to
provide close to acceptable fits. The Teff = 8750 K and log g = 4.0 model was found
to give a visibly better fit than other models tested, and hence has been adopted
as the best fit temperature. The difference in fits between the initial and best fit
models is illustrated in Figure 3.2. The Teff = 9000 K and log g = 4.0 model was
only marginally worse than the best fit case, and cannot be completely ruled out.
Additionally, the sensitivity of the fit to log g is weaker than the the sensitivity to
temperature. However, log g = 4.5 did produce several small discrepancies that were
corrected by a decrease to log g = 4.0, thus we prefer the Teff = 8750 K and log g
= 4.0 model.
3.4
Least Squares Deconvolution
For the typical field strength and rotation velocity observed for magnetic intermediate
mass stars (∼1 kG and ∼40 km s−1 ), the circular polarization signal due to a magnetic
field in any given line in an ESPaSOnS spectrum is very small. In the case of HD
72106A, which has a field strength typical of Ap/Bp stars, the amplitude of the Stokes
V signal is about 0.1-0.3 percent of the Stokes I continuum level (a typical absorption
line in Stokes I has a depth of ∼ 10% of the continuum). Consequently, a technique
90
CHAPTER 3. ANALYSIS
that can effectively average over many lines in a spectrum to produce a high signalto-noise ratio would be very valuable. Least Squares Deconvolution (LSD) does this
(Donati et al., 1997). This averaging is possible because most metallic lines in a
spectrum have approximately the same Stokes I and V profile shape. The spectrum
of a star can very approximately be described as a convolution of an average line
profile and a set of delta functions, located at each individual line’s wavelength and
with an amplitude equal to the line’s depth. The objective of LSD is to reverse this
process and deconvolve the average line profile from the observed spectrum, using
a theoretical set of delta functions (referred to as a line mask). The details of the
procedure are outlined here for the Stokes V spectrum, although the procedure can
also be applied to Stokes I, Q, and U with minimal modification (Wade et al., 2000a)
In the weak-field regime (when magnetic line splitting is much less than thermal
Doppler broadening), the Stokes V profile of a line, local to one point on a star’s
surface, follows the relation (Landstreet, 1982):
Vloc (v) ∝ gλ
∂Iloc (v)
Bz ,
∂v
(3.9)
where Iloc (v) is the local Stokes I profile of the line, g is the Landé factor (magnetic
sensitivity), Bz is the line-of-sight (longitudinal) magnetic field strength, and v is the
velocity coordinate c∆λ/λ (with ∆λ being the wavelength shift relative to the center
of the line at wavelength λ) (Donati et al., 1997). The proportionality constant is
the same for all lines. If one assumes that Iloc (v) has the same shape for all lines and
simply scales with line depth d then one can write:
Vloc (v) = gλdkB (v),
(3.10)
where kB (v) is a proportionality function, the same for all lines, which includes the
effect of the longitudinal field. This assumption is strictly true only for optically
91
CHAPTER 3. ANALYSIS
thin lines. However, for profiles dominated by rotational broadening it is not a bad
approximation (Donati et al., 1997). Integrating Vloc (v) over the surface of a star can
effectively be done by integrating over M points in brightness bM and radial velocity
vM to produce the global V (v). Brightness and radial velocity are used since they
provide simpler coordinates for local profiles, which are generated at one particular
radial velocity and brightness from the point of view of the observer. The global V (v)
can be written as:
V (v) =
ZZ
bM Vloc (v − vM ) dS
ZZ
= gλd
bM kB (v − vM ) dS
(3.11)
= w Z(v).
If limb darkening is assumed to be constant for all wavelengths, then Z(v), the Zeeman
signature (or LSD profile), is constant for all lines. In that case the shape of the
Zeeman signature is reproduced by all Stokes V profiles (V (v)) scaled by the factor
w = gλd (Donati et al., 1997).
The line mask can be represented as:
M (v) =
X
wi δ(v − vi ),
(3.12)
i
where vi is the position, in velocity units, of the ith line, wi is the weighting (depth)
of that line, and δ is the Dirac delta function (Donati et al., 1997). This effectively
gives a list of center wavelengths and weightings for all lines of interest. The Stokes
V spectrum can then be described as the convolution of Z(v) and M (v) (that is
V = M ∗ Z), or in terms of a linear system: V = M · Z. This assumes that
line intensities add linearly, which is clearly not true, especially for stronger, more
saturated lines. In intermediate mass stars with moderate rotation rates, such as HD
CHAPTER 3. ANALYSIS
92
72106 A and B, there are many unblended or weakly blended lines available in the
spectrum, lessening this concern somewhat. In practical experiments, Donati et al.
(1997) found that good results were achieved despite this assumption. Other authors,
for example Wade et al. (2000a) and Shorlin et al. (2002), support this conclusion.
One can then seek the best fit least-squares solution by minimizing the χ2 function
(Press et al., 1992):
χ2 = t (M · Z − V) · S2 · (M · Z − V),
(3.13)
with Z (the Zeeman signature) as the matrix of free parameters and the observed
Stokes V spectrum (V). The matrix S is a square diagonal matrix whose elements
Sjj are the inverse error bars (1/σj ) of the j th pixel in the spectrum. Since our model
is linear, the χ2 problem can be solved using the method of normal equations (e.g.
Press et al., 1992) (essentially setting the derivative of χ2 to 0). This gives the result:
(t M · S2 · M) · Z = t M · S2 · V,
(3.14)
Z = (t M · S2 · M)−1 t M · S2 · V.
(3.15)
or equivalently:
The term t M · S2 · V is the cross correlation of the observed spectrum (V) with
the line mask (M). Uncertainties can be estimated from the diagonal elements of
the t M · S2 · M matrix (the auto-correlation matrix). In the case when blending of
lines is minimal and photon noise dominates uncertainties, as is often seen in Stokes V
spectra, this gives a good estimate of the real uncertainty (Donati et al., 1997). Cases
when photon noise is small and improperly calculated blends and other systematic
limitations of the model become significant can be diagnosed by large values of χ2 .
In these cases, photon noise based error bars systematically underestimate the true
93
CHAPTER 3. ANALYSIS
uncertainty. Therefore the photon noise error bars are scaled up to compensate for
the large χ2 values, in the manner described by Wade et al. (2000a).
LSD analysis was performed on all spectra obtained. Observations of the combined
system were transformed into reconstructed spectra of the primary before LSD was
performed. A line mask tailored to a star with Teff = 10000 K and Ap-like overabundances (see Shorlin et al., 2002) was used for the primary. An 8000 K line mask
with solar abundances was used for the secondary. Sensitivity to line mask details
is small, thus discrepancies in temperature or abundance of line masks have little
impact on the resulting profiles. Sample LSD profiles for both components are shown
in Figure 3.7.
B
A
1.03
25 x V/I
25 x V/I
I
I
1.02
Flux
1.01
1
0.99
0.98
0.97
-100
-50
0
50
Velocity (km/s)
100
150
-100
-50
0
50
Velocity (km/s)
100
150
Figure 3.7: Sample Stokes I and V LSD profiles for HD 72106A (in the left frame,
labeled A) and HD 72106B (in the right frame, labeled B). A Zeeman signature,
indicating the presence of a magnetic field, is clearly visible in frame A, but absent in
frame B. A line asymmetry is visible in the Stokes I profile in frame A, due to surface
abundance non-uniformities.
CHAPTER 3. ANALYSIS
3.5
94
Magnetic Field Analysis
Analysis of a single Stokes V profile can be used to determine the mean longitudinal
magnetic field (the surface component of the magnetic field, projected onto the line
of sight, and integrated over the visible disk of the star) at one point in time (phase).
For our purposes, we wish to calculate the longitudinal field from the LSD profiles
discussed in the previous section. This is necessary, as the original spectra have
too poor a S/N in individual lines to determine precise magnetic field strengths. The
mean longitudinal field strength (hBz i) can be calculated from the first-order moment
of the LSD Stokes V profile (Mathys, 1989) as follows:
R
vV (v)dv
11
R
hBz i = −2.14 × 10
,
λgc [Ic − I(v)]dv
(3.16)
where g and λ (in nm) are respectively the average Landé factor and wavelength
for all lines used in the LSD mask (Donati et al., 1997; Wade et al., 2000b). In
equation 3.16 c is the speed of light, v is the wavelength in velocity units, I(v) and
V (v) are the Stokes I and V LSD profiles, and Ic is the Stokes I continuum level
(which should be unity for a well normalized spectrum). For the primary the values
g = 1.27 and λ = 527 nm were used, for the secondary g = 1.21 and λ = 535 nm
were used. Longitudinal magnetic field measurements, obtained using equation 3.16,
are presented in Table 3.2. Integration was performed through the portion of the
line profile that exceeded 15% of the total line depth, with an additional 5 km s−1
on either end. This was done to avoid integration over the continuum, ‘diluting’ the
longitudinal magnetic field’s signal by increasing the noise. The continuum level Ic
is taken from the average value of the Stokes I LSD profile far from the absorption
line. The integration (in the velocity coordinate) is performed with respect to the
95
CHAPTER 3. ANALYSIS
‘center-of-gravity’ (cog) of the line. The center-of-gravity is calculated by evaluating
the expression:
R
v|V (v) (Ic − I(v)) |dv
.
cog = R
|V (v) (Ic − I(v)) |dv
(3.17)
Using the center-of-gravity provides an accurate value for the velocity at the center
of the line, even for complicated asymmetric line profiles. This allows one to correct
for the (usually unknown) Doppler shift of the star. Uncertainties are determined by
propagating the error bars on the LSD profile through the numerical integration and
arithmetic of equation 3.16.
Longitudinal magnetic fields are detected, at 3σ or better, in nearly all our observations of the primary, but no longitudinal field is detected in the secondary. Additionally, the Stokes V profiles of HD 72106B are consistent with a null field hypothesis.
Wade et al. (2007) find no magnetic field in HD 71206B with their FORS1 observation.
They report longitudinal magnetic fields of 52 ± 90 G from an analysis of Balmer
lines and 3 ± 122 G from an analysis of metallic lines. This result is consistent with
the earlier analysis of HD 72106B by Wade et al. (2005), in which no magnetic field
was detected.
One can constrain the geometry of a stellar magnetic field using a series of longitudinal field observations with known rotation phases. In the case of a tilted, centered
dipole field, the longitudinal field will vary sinusoidally with the rotation period of the
star (Prot ). Models of this type are known as oblique dipole rotators. The well-known
relation of Preston (1967) describes the variation of the mean longitudinal magnetic
field hBz i as a function of rotation phase φ. The rotation phase is given by:
φ=
t − t0
,
Prot
(3.18)
where t is the time of interest and t0 is some reference time. For simplicity, values
96
CHAPTER 3. ANALYSIS
UT Date
22 Feb.
09 Jan.
11 Jan.
12 Jan.
11 Feb.
11 Feb.
12 Feb.
13 Feb.
13 Feb.
02 Mar.
02 Mar.
02 Mar.
03 Mar.
04 Mar.
05 Mar.
05 Mar.
09 Mar.
09 Mar.
12 Jan.
05 Mar.
05
06
06
06
06
06
06
06
06
07
07
07
07
07
07
07
07
07
06
07
HJD
Component
(-2 450 000)
3423.9248
3745.02967
3747.02034
3747.99629
3777.87860
3777.95149
3778.86172
3779.87202
3779.98127
4161.77282
4161.80256
4161.90282
4162.85713
4163.83561
4164.88387
4164.90961
4168.85791
4168.90947
3748.01496
4164.84650
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
B
B
Bz
(G)
Significance
of Bz
LSD
Detection
300 ± 60
410 ± 50
320 ± 80
-10 ± 40
200 ± 50
90 ± 40
260 ± 40
140 ± 60
120 ± 170
330 ± 40
270 ± 40
190 ± 40
230 ± 50
180 ± 30
340 ± 30
350 ± 40
280 ± 40
230 ± 40
0 ± 170
-50 ± 60
5.3
9.1
3.8
0.2
4.4
2.2
6.6
2.4
0.7
8.1
7.1
5.3
4.7
5.6
13.5
9.3
7.6
5.8
0.0
0.9
D
D
D
D
D
D
D
D
N
D
D
D
D
D
D
D
D
D
N
N
Table 3.2: Mean longitudinal field measurements for each observation of HD 72106A
and B. The significance of Bz is the number of standard deviations that the detection
corresponds to, we consider 3σ necessary for a confident detection. The letter D in
the LSD detection column indicates that the Stokes V LSD profile is inconsistent with
a flat profile at the 99.99% confidence limit, as determined by its χ2 statistic. This
indicates the presence of a magnetic field, even if the mean longitudinal component
not clearly detected. The letter N indicates that the inconsistency was at the 99.9%
limit or less, and hence cannot be take as strong evidence for the detection of a
magnetic field (Donati et al., 1997).
97
CHAPTER 3. ANALYSIS
of φ between 0 and 1 are often used. One can then write the variation of the mean
longitudinal magnetic field as
hBz i = Bp
15 + u
(cos β cos i + sin β sin i cos 2πφ),
20(3 − u)
(3.19)
where Bp is the strength of the dipole magnetic field at the magnetic pole, u is the
limb darkening parameter, i is the inclination of the rotation axis of the star to our
line of sight, and β is the angle between the rotation axis and magnetic dipole’s axis
(the obliquity angle). For equation 3.19, the t0 term in φ (see equation 3.18) must be
set such that the maximum longitudinal field strength is observed when φ = 0. The
angles i and β are related by (Preston, 1967):
tan β =
where r =
Bzmin
Bzmax
1−r
cot i,
1+r
(3.20)
is the ratio of the smallest (Bzmin ) to the largest (Bzmax ) absolute
longitudinal field strengths from the complete sinusoidal curve. Determining i is
discussed in Sections 3.6 and 4.2. Thus if one has a good set of observations describing
the sinusoidal variability of the longitudinal magnetic field in a star, the dipole field
parameters (Bp and β) can be calculated straightforwardly, with a priori knowledge
of i. This analysis was performed for HD 72106A, but not for HD 72106B because
we do not detect a magnetic field in the secondary.
As seen in Table 3.2, a Stokes V signature from a magnetic field can often be
detected when a mean longitudinal field cannot. In an effort to verify the validity of
the dipole magnetic field model, and possibly to constrain the field geometry further,
direct modeling of the Stokes V LSD profiles was undertaken.
Modeling of the LSD Stokes V profiles of HD 72106A was performed with the
ZEEMAN2 spectrum synthesis code, described in Section 3.3. This process was not
CHAPTER 3. ANALYSIS
98
performed for HD 72106B since there was no evidence for a magnetic field in that star.
Mean atomic line data from the LSD process were used. Chemical abundance, and
hence line depth, for our synthetic LSD line was fixed by fitting the equivalent width
of the Stokes I LSD profile. For the magnetic field geometry, initially the oblique
dipole rotator model derived from the longitudinal magnetic field measurements was
used. Comparison of the set of phased Stokes V LSD profiles with model profiles
at corresponding phases showed acceptable agreement, as discussed in Section 4.3.
This suggests that a dipole field is a sufficient model, within the uncertainty of our
observations.
A grid of dipole models was constructed by generating many synthetic time series
of Stokes V profiles by varying β and Bp . Model Stokes V profiles were calculated for
the same rotational phases as the observations. Initially a grid spacing of 100 G and
10 degrees was used; this was refined to 20 G and 1 degree near the best fit model.
The reduced χ2 of the model3 , including all points in the line profiles (and excluding
points in continuum) from all rotation phases, was calculated for each point in β and
Bp , producing a map of reduced χ2 . The best fit model was then simply selected by
finding the minimum χ2 value. An illustrative map of χ2 for β and Bp in the star
θ 1 Ori C (adapted from Wade et al., 2006a) is shown in Figure 3.8. When the range
of models were examined by eye, significant departures from the best fit model (by a
few hundreds of gauss or a few tens of degrees) showed clear discrepancies between
modeled and observed profiles at several phases.
One can use the change in χ2 away from the minimum in χ2 to place limits,
at a specific confidence level, on a model. This is described by Press et al. (1992,
3
Reduced χ2 is simply the standard χ2 value divided by the number of ‘degrees of freedom’ (the
number of observed data points minus the number of fit parameters)
99
CHAPTER 3. ANALYSIS
Figure 3.8: A sample χ2 map in Bp and β for θ 1 Ori C, adapted from Wade et
al. (2006a). The gray-scale axis represents reduced χ2 . Values outside the 95%
confidence limit are white. Synthetic model profiles, using an inclination angle of
45◦ , were compared to observed LSD Stokes V profiles to produce this plot.
Section 15.6) and was originally developed by Avni (1976) and Lampton et al. (1976).
Based on the number of free parameters in the model, one can used χ2 statistics to
calculate the change in χ2 that corresponds to a specific confidence limit. The change
in χ2 (∆χ2 ) can be determined by finding the point in the cumulative χ2 distribution
function, with the degrees of freedom equal to the number of free model parameters,
that has a probability equal to the desired confidence limit. The equation governing
this is (Press et al., 1992):
P
ν ∆χ2
,
2 2
≡
γ(ν/2, ∆χ2 /2)
= p,
Γ(ν/2)
(3.21)
where P is the ‘regularized’ gamma function, γ is the lower incomplete gamma function, Γ is the ‘complete’ gamma function, ν is the number of free model parameters,
and p is the desired confidence limit. Thus for a given p one must solve for ∆χ2 . For
CHAPTER 3. ANALYSIS
100
example, with 2 free parameters at a 99.73% confidence limit (∼ 3σ) one requires a
change in χ2 of 11.8. The limiting value of χ2 (χ2lim ), at confidence p, is then simply
the sum of the minimum χ2 (χ2min ) and ∆χ2 , that is χ2lim = χ2min +∆χ2 . The produces
a boundary in parameter space that can be used to determine error bars. Figure 3.8
provides an illustration of such a boundary in two dimensions. The error bars (i.e.
confidence intervals for individual parameters) are, in general, the projection of the
extrema of the confidence boundary onto the relevant coordinate axis.
This method of determining uncertainties for our model was used, providing much
more rigorous error bars than the ‘by eye’ method. To properly include the substantial
uncertainty in inclination angle, we recalculate the grid of models at the maximum
and minimum values of i allowed by its error bars. Our final uncertainties then run
from the 3σ limit at one inclination angle extreme to the 3σ limit at the other inclination angle extreme. The results from this are discussed in Section 4.3. The greatest
limitation in these models is the inaccurately modeled Stokes I profiles. Equivalent
width was accurately modeled, however line asymmetries due to surface abundance
inhomogeneities were not modeled. As a consequence of the unmodeled surface abundance inhomogeneities, there was a limitation on the accuracy of the Stokes V profile
shapes (which are also affected by the presence of abundance spots). Hence the true
errors bars on the magnetic geometry may be larger than the formal uncertainties
derived above. The technique outlined above has been employed by a number of
authors (e.g. Donati et al., 2001; Wade et al., 2006a) with good results.
CHAPTER 3. ANALYSIS
3.6
101
Period Determination
In order to determine the inclination angle i of HD 72106A, it was necessary to
determine the rotation period of the star. Additionally, to determine the phase of
each observation, necessary for magnetic field modeling or Doppler Imaging, a rotation
period is required.
In order to determine the rotation period of a star, one needs an observable property of the star that varies with the star’s rotation. In a normal main sequence
A or B star such a property often does not exist. However, in the case of Ap/Bp
stars, and in the case of HD 72106A, we have a detectable longitudinal magnetic
field with presumably periodic variability, as well as presumably periodic variability
in the line profiles of the star. The periodicity (illustrated by the similarities between
some profiles separated by two nights), and stability between cycles (illustrated by
the similarities between some profiles separated by over a year), of the magnetic field
and line variability (in absorption) strongly suggests that they are intrinsic to the
star (e.g. Preston, 1967; Hatzes et al., 1989), thus we adopt an oblique rotator model.
That is, simply put, that the property in question varies across the surface of the
star. As the star rotates at some inclination angle, periodic variability is observed
due to different parts of the stellar surface being visible at different times.
The simplest data-set to investigate is the set of longitudinal magnetic field measurements. The variability of a centered dipole magnetic field should be a first-order
sinusoid. If the magnetic field is a low-order multipole, then the observed longitudinal
field varies as a low-order sinusoid.
In order to find the best fit rotation period, a periodogram (a graph of χ2 versus period) was constructed for a first-order sinusoid fitting function. A first-order
102
CHAPTER 3. ANALYSIS
sinusoid with a fixed period was fit through the longitudinal field observations and
reduced χ2 calculated. The phase, amplitude and average value of the sinusoid were
taken as free parameters. This was repeated for many period values, with a logarithmic sampling from 0.3 to 10 days, to produce a graph of reduced χ2 versus period.
One cannot simply use a χ2 minimization routine to find the best period because
there are a great many local minima in the parameter space. Any reasonably efficient
χ2 minimization routine would find it almost impossible to reliably find the global χ2
minimum. The resulting periodogram, based on longitudinal field measurements, is
shown in Figure 3.9. The procedure described here is similar to performing a Fourier
analysis on the set of longitudinal field measurements and dates, however there are a
few differences (Press et al., 1992).
This method of fitting sinusoids through the data points is essentially a LombScargle method (Press et al., 1992). This method has the advantages of working well
with unevenly sampled data and being sensitive much below a Nyquist frequency
calculated from the average data point spacing. The Nyquist frequency, fc , is defined
by (Press et al., 1992):
fc =
1
2∆
(3.22)
where ∆ is the spacing in time between data points. In a uniformly sampled data set,
frequency components above the Nyquist frequency are falsely shifted into the range
below the Nyquist frequency, a process known as aliasing (Press et al., 1992). Thus
in a search for periodicity, if the period is shorter then the Nyquist period one can
see false peaks in a power spectrum. In the case of a rotating star, the star would
be completing multiple rotations between observations. For non-uniformly sampled
data, periodicity can be found via the Lomb-Scargle method, much below the Nyquist
103
CHAPTER 3. ANALYSIS
7
6
5
Reduced χ
2
4
3
2
1
0
0.5
1
1.5
2
2.5
Period (Days)
3
3.5
4
Figure 3.9: Periodogram for HD 72106A, using a first-order sinusoid, based on longitudinal field data. A number of strong minima are apparent between 0.4 and 2
days, with the ∼ 0.64 day period giving the strongest minimum. The horizontal line
represents the 99% confidence limit, based on the minimum at ∼ 0.64 days.
CHAPTER 3. ANALYSIS
104
frequency from the average spacing of data points (Press et al., 1992). Since some
data points are more closely spaced than the average, they provide constraints below
the Nyquist frequency based on the average spacing. In principle one can determine
a period down to nearly two times the smallest data spacing, but in practice, with
limited noisy data sets and a limited number of points, this is overly optimistic. With
this advantage comes two drawbacks. The Lomb-Scargle method is computationally
much slower than Fourier methods (which can make efficient use of Fast Fourier
Transforms; Press et al., 1992). More importantly, the significance of periods found
with the Lomb-Scargle method depends on the number of data points used (Press
et al., 1992), thus a large number of observations is necessary for accurate period
determination.
The dependence of the Lomb-Scargle method’s accuracy on the number of observations proved to be problematic for us. Specifically, we found many minima with
almost identical reduced χ2 in the periodogram. One can assign confidence limits in
terms of χ2 (or reduced χ2 ), using the minimum χ2 value and the number of free
fitting parameters, as described by Press et al. (1992, Section 15.6) and discussed
near the end of Section 3.5. However, in the case of the longitudinal field data we
could not draw any useful conclusions, at the 99% confidence level, since there were
several periods with suitable reduced χ2 values.
A full Stokes V LSD profile contains significantly more information than the corresponding single derived longitudinal field datum. In light of this, we hoped to
improve the periodogram analysis by searching for periodic variability in the Stokes
V and Stokes I LSD profiles. The method developed for this is based on the fitting
of sinusoids as described above. However, rather than fitting the sinusoid through
CHAPTER 3. ANALYSIS
105
a time series of longitudinal field measurements, it is fit through the times series of
observations at one particular point (‘pixel’) in the LSD profile. Again a wide range of
periods are used, creating a periodogram for one particular pixel. The phase, amplitude and average value of the sinusoid were used as free parameters for the fit, and the
periods considered were logarithmically sampled from 0.3 to 10 days. Periodograms
are created in this fashion independently for all the pixels across the LSD profile.
Then, so as to improve the S/N, the periodograms for all of the pixels are averaged
together, weighted by the significance (in standard deviations) of the variability in
each pixel. This produces an average periodogram for the LSD profile. This procedure is performed independently for the Stokes I and Stokes V LSD profiles. The
method described here was in part inspired by Adelman et al. (2002) and described
briefly by Aurière at al. (2007).
The technique was tested on a number of ‘normal’ main sequence Ap and Bp
stars, with good results (Aurière at al., 2007; Power, 2007). Additional tests using
synthetic I and V profiles with artificial noise and realistically sparse phase coverage
were also performed. In these tests the true period was identified unambiguously, with
no other minima below the 99% cutoff level. Stokes I and V based periodograms have
produced unambiguous periods from a number of observations for which longitudinal
field based periodograms were insufficient (Aurière at al., 2007; Power, 2007). The
technique performs quite well for Stokes V profiles, in which approximately sinusoidal
variability in individual pixels is expected for a low-order multipole magnetic field.
Stokes I profile based periodograms perform well at identifying the proper period, but
generally have large χ2 values. A pixel in a Stokes I profile will not, in general, vary
in a sinusoidal fashion. However, the fitting of a sinusoid picks out periodicity well.
CHAPTER 3. ANALYSIS
106
Thus one sees large χ2 values, but reliable minima in the periodogram. The LSD pixel
average periodograms described here are equivalent to averaging together Fourier
transforms of the data for each individual pixel in the LSD profile. Periodograms
based on LSD Stokes I and V profiles, using a first-order sinusoid, for HD 72106A
can be seen in Figure 3.10.
Despite this more advanced technique, some ambiguity in the rotation period of
HD 72106A remained. Several candidate periods with strong minima in all periodograms still could not be ruled out at the 99% confidence limit. These periods
were investigated for non-physical variations of their LSD profiles, as detailed in Section 4.2. Dramatic changes in profile shape with correspondingly small changes in
rotation phase, or features moving counter to the Doppler motion expected due to
stellar rotation (from red to blue) were considered grounds for rejecting a period as
unphysical. This was finally sufficient to eliminate all but one period, 0.63995 days,
as discussed in Section 4.2.
3.7
Doppler Imaging
Doppler Imaging is a technique for indirectly reconstructing surface features of a star
by using a series of spectra obtained at different rotation phases, and exploiting the
rotational Doppler effect. As the star rotates, a surface feature, for example a spot
of chemical overabundance, moves across the visible disk of the star. This change
is reflected in a rotationally broadened line profile, as different points on the star’s
surface have different radial velocities as seen by an observer, and hence different
Doppler shifts. Thus the rotational longitude of a surface feature can be deduced
from the corresponding spectral feature’s position in a rotationally broadened line
107
CHAPTER 3. ANALYSIS
Reduced χ
2
10
8
6
4
2
0
Reduced χ
2
10
8
6
4
2
0
0.5
1
1.5
2
2.5
Period (days)
3
3.5
4
Figure 3.10: Periodogram for HD 72106A, using a first-order sinusoid, based on the
LSD Stokes I (top frame) and Stokes V (bottom frame) variations. The horizontal
lines represents the 99% confidence limits, based on the deepest minimum in each
plot. A number of comparable minima are still present, however the scaling in χ2 is
larger, and fewer minima fall below the 99% confidence limit.
CHAPTER 3. ANALYSIS
108
profile. Latitudinal information is contained in the variation with phase of a feature
in a line profile. A feature near the rotational pole would present smaller Doppler
shifts, move more slowly across the line profile, and be visible longer.
Figure 3.11: Illustration of Doppler Imaging. As the model star rotates, the spot
of under-abundance moves across the visible disk of the star. The ‘bump’ in the
synthetic line profile, corresponding to the spot of under-abundance, moves across
the line profile as the star rotates, due to the rotational Doppler shift. In the case
in which the spot is at higher latitude, (case A rather than case B) the ‘bump’ is
restricted to the center of the line profile, since it never achieves as large a Doppler
shift as the equator. Image: A. P. Hatzes, Thüringer Landessternwarte Tautenburg
(http://www.tls-tautenburg.de/research/artie/di technique.html).
In Doppler Imaging, the information contained in the time series of variable line
profiles is extracted by inverting the line profiles to produce a surface map of the star.
In the case of abundance Doppler Imaging this is, in essence, done by fitting the set
of unknown abundances across the surface of the star to the observed series of line
profiles. At the rotation phase of each observation, a synthetic line profile can be
109
CHAPTER 3. ANALYSIS
calculated from an assumed model surface abundance distribution. By varying the
surface abundance distribution one can then fit the synthetic line to the observed line
profile. This is done simultaneously for all observed rotation phases to produce the
best fit surface abundance map.
In mathematical terms, this inverse problem is solved by minimizing the quantity
(e.g. Kochukhov et al., 2004):
E=
XX
φ
2
[Icalc (λ, φ) − Iobs (λ, φ)]2 /σobs
(λ, φ) + R(ε)
λ
(3.23)
= χ2 + R(ε),
where Icalc (λ, φ) and Iobs (λ, φ) are the calculated and observed line profiles at wave2
length λ and phase φ, and σobs
(λ, φ) is the uncertainty associated with the observed
profile at wavelength λ. The sums over phases (φ) and wavelengths (λ) cover all
observed data points, and thus summation yields χ2 . The last term in equation 3.23,
R(ε), is the regularization function, and warrants further discussion.
In general, the inversion required for Doppler Imaging is an ‘ill-posed’ problem.
The large number of surface elements necessary for a map with reasonable resolution
produces several thousand free parameters, which are not uniquely constrained by
the relatively small number of observed data points4 . In principle there can be many
solutions which provide equally good fits to the observed line profiles. Thus a unique,
stable solution cannot be achieved without the addition of some further constraint:
the regularization function. Regularization provides a scheme for selecting a particular solution from a number of possible solutions that fit the data. Most possible
solutions present large variations between nearby points on the stellar surface. These
4
In the case of our observations we have approximately 20 points across a line profile and observations at 18 phases, hence 360 data points.
110
CHAPTER 3. ANALYSIS
solutions are usually non-physical, and contain variations below the true resolution
of the Doppler map. Thus most regularization schemes search for the most smooth
or uniform solution that fits the observations. A number of regularization schemes
for Doppler Imaging exist (e.g. Vogt et al., 1987; Piskunov et al., 1990), however the
differences are minimal when the line profile variability is large compared to the noise
in the observation (Piskunov et al., 1990; Korhonen et al., 1999).
In this thesis we used the the Doppler Imaging code INVERS12, developed by
O. Kochukhov and N. Piskunov (Kochukhov et al., 2004), for surface chemical abundance mapping. This program performs accurate LTE spectrum synthesis, using
pre-calculated model stellar atmospheres. INVERS12 allows for simultaneous modeling of multiple chemical elements and multiple wavelength regions, and takes into
account blended lines. Tikhonov regularization is used, which has the functional form
(Tikhonov, 1963; Kochukhov et al., 2004):
ZZ
R(ε) =
||∇ε(M )||2 dM,
(3.24)
where ε(M ) is the surface distribution of the relevant element, the integration is
performed over latitude and longitude (i.e. M ), and || indicates the absolute value of
a vector. This regularization essentially constrains the reconstructed abundances to
change slowly over the surface of the star. Thus Tikhonov regularization selects the
‘smoothest’ solution that still fits the data.
The disk integrated synthetic line profiles are calculated from local synthetic line
profiles, integrated across the visible surface of the star. Surface features are then
included by their effect on the local line profiles. Mathematically this can be written
as (Kochukhov et al., 2004):
Icalc (λ, φ) =
RR
IL (M, ε, λ + ∆λD (M, φ)) cos θdM
RR
,
IC (M ) cos θdM
(3.25)
CHAPTER 3. ANALYSIS
111
where IL and IC are the line and continuum intensities, the latter assumed to be
independent of the local chemical composition. Division is necessary to produce a
continuum normalized line profile. M is the position on the stellar surface in longitude
and latitude, θ is the angle between the line of sight and the normal to the surface at
point M , and ∆λD (M, φ) is the Doppler shift of the surface element M at phase φ.
Integration is performed over the hemisphere of the star visible at rotation phase φ.
For input into the Doppler Imaging computations for HD 72106A we used the
adopted effective temperature and surface gravity from Section 3.1.1: Teff = 11000 K
and log g = 4.0. An initial v sin i of 41 km s−1 and abundances, for the treatment of
blended lines, were taken from the results of our abundance analysis of HD 72106A,
presented in Section 4.1.1. A further discussion of the input parameters can be
found in Section 4.4. Initially, five Si II lines were used simultaneously to construct a
Doppler Imaging map of Si across the surface of HD 72106A. However, the noise level
in the observed spectra proved to be too high for very accurate Doppler Imaging. To
remedy this, we performed Doppler Imaging using LSD profiles, constructed using
lines of individual elements. LSD produces line profiles with a much higher S/N than
individual spectral lines, which can be used much like regular spectral line profiles
(e.g. Barnes et al., 1998; Donati et al., 2000; Marsden et al., 2005). Ultimately, high
quality maps were constructed for Si, Ti, Cr, and Fe from LSD profiles. These results
are presented in Section 4.4, along with a further discussion of the challenges faced.
The identification of individual lines for Doppler Imaging, preparation of those
lines for analysis, construction of LSD profiles, the determination of fundamental stellar parameters, and determination of initial parameters for Doppler Imaging were all
performed by the author. The execution of the Doppler Imaging code was performed
CHAPTER 3. ANALYSIS
112
by Dr. O. Kochukhov. Discussions with Dr. Kochukhov about the interpretation of
the Doppler Imaging results were invaluable.
Chapter 4
Results
4.1
Surface Chemical Abundances
Detailed chemical abundance analyses were performed for both components of HD
72106, simultaneously fitting v sin i, surface chemical abundances for many species,
and microturbulence (for HD 72106B). The abundances of HD 72106A represent mean
surface abundances, as there exists strong evidence for the presence of surface abundance inhomogeneities. For the primary, abundances for 13 elements in 7 independent
spectral windows were modeled. For the secondary, abundances for 15 elements in 7
windows were obtained.
4.1.1
Primary
Clear chemical peculiarities in HD 72106A, as hinted at by the observed line profile
variability, were discovered early in the spectrum modeling process. In particular, Fe
over-abundances with respect to the solar abundance of +1 dex (1 dex = 1 order of
113
CHAPTER 4. RESULTS
114
magnitude), Cr over-abundances of +2 dex, and Nd over-abundances of about +3
dex were obtained. When one considers that these values are in logarithmic units,
it is apparent that the departures from solar abundances in the atmosphere of HD
72106A are quite strong. The abundance patterns observed are characteristic of main
sequence Ap and Bp stars, although remarkably strong.
HD 72106A was modeled using an Atlas9 model atmosphere with a temperature
of 11000 K and log g = 4.0. A 1 kG dipole magnetic field was included, with an
obliquity angle β = 90◦ . While later results show this magnetic field geometry is
not perfectly correct, it provides a good approximation for the magnetic desaturation
which will affect the Stokes I profiles. A modification of the field geometry to Bp =
1300 G and β = 60◦ (the final adopted geometry) has a negligible impact on the
results, well below the level of the uncertainties. No microturbulence was included,
as magnetic intermediate mass stars (particularly Ap and Bp stars) do not display
any evidence of microturbulence (e.g. Ryabchikova et al., 2000). The strong, globally
ordered magnetic field seen in these stars is sufficient to suppress microturbulence
(e.g. Ryabchikova et al., 2000). Moreover, no evidence for microturbulence was found
during the modeling process. Fitting the radial velocity of the star, by adjusting the
net Doppler shift of the observed spectrum, produced a value of 22 ± 1 km s−1 . As
described in Section 3.3.2, individual regions ranging from ∼ 100 Å to ∼ 200 Å of
the observed spectrum were fit independently. The observation of the spectrum of
HD 72106A on January 12, 2006 was used for fitting. As discussed in Section 3.3.2,
this spectrum is representative of the average spectrum (over all observations) of the
star.
Results for each individual segment of spectrum fit can be seen in Table 4.1.
115
CHAPTER 4. RESULTS
Abundances are presented in units of log
Nel
,
NH
where Nel is the number of atoms of the
element in question and NH is the number of H atoms present. Spectrum-averaged
abundances are presented numerically in Table 4.2, and graphically in Figure 4.2.
The spectrum averaged values should be taken as the most accurate abundances.
Example segments of fit spectrum can be seen in Figure 4.1.
As discussed in Section 3.3.2, uncertainties on the spectrum-averaged, best fit
values are based on the standard deviation of the values from individual windows.
Exceptions to this rule are marked with an * in Table 4.2. In these cases only a
few lines were available for modeling, so the uncertainties were estimated by taking
the change in abundance necessary to shift the synthetic line well beyond any noise
or normalization errors in the observed spectrum. Thus the uncertainties in these
special cases represent the ∼ 2σ confidence level, while all other uncertainties are at
1σ.
Remarkably strong over-abundances of Cr, Fe and Nd are found. The Si abundance appears to be above solar by 0.8 dex, whereas He appears to be ∼ 1.5 dex
under-abundant. A number of elements, such as Al, Sc and Sr, appear to have solar
abundances. A couple of elements, particularly Mg and O, hint at possible peculiarity
but require further study before concrete conclusions can be drawn. The v sin i value
of 41.0 ± 0.7 km s−1 is very low for a main sequence B star, but within the normal
range of values for a Bp star. The strong over-abundances in Si, Cr, Fe, and Nd, as
well as the under-abundance in He, are common features of cooler Bp stars (Jaschek
& Jaschek, 1995).
Strong chemical peculiarities, particularly in He, and magnetic fields can affect
the structure of a stellar model atmosphere, which can in turn modify the derived
He
O
Mg
Al
Si
Ca
Sc
Ti
Cr
Fe
Sr
Ba
Nd
4400-4500 Å
4500-4600 Å
4600-4700 Å
5000-5200 Å
5200-5400 Å
5400-5600 Å
41.4
40.1
41.0
41.0
41.9
40.1
41.4
-3.6
-3.0
-4.5
-3.9
-2.8
-4.1
-3.5
-5.1
-8.6
-6.0
-4.4
-3.5
-8.7
-6.2
-4.5
-3.5
-6.0
-4.2
-3.6
-5.8
-3.5
-4.1
-4.4
-3.5
-5.9
-4.4
-3.5
-6.1
-4.3
-3.5
-7.4
-7.6
CHAPTER 4. RESULTS
v sin i (km s−1 )
4170-4265 Å
-3.8
-4.1
-3.4
-8.5
Table 4.1: Chemical abundances for HD 72106A in each independently fit segment of spectrum. Clear overabundances in a number of elements can be seen.
116
117
CHAPTER 4. RESULTS
1
Fe Fe
Ti
Si
Flux
0.9
0.8
Fe
Fe
Ti
Ti
Fe
He Fe Fe
Cr
Fe
Cr
Ti
0.7
0.6
Fe
Mg
4450
4470
Wavelength (angstroms)
4460
4480
4490
1
Flux
0.95
0.9
0.85
0.8
5000
Fe
Fe
Cr Fe
Fe
Fe
Si
Fe
Fe
Fe
5010
5020
5030
Wavelength (angstroms)
Fe
Fe
Fe
Si
5040
5050
Figure 4.1: Sample best fit synthetic spectra for HD 72106A in two independently
fit spectral windows. Major contributors to each line have been labeled, in order of
importance. The smooth solid line is the best fit spectrum in this region, the dashed
line is a spectrum computed with solar chemical abundances.
118
CHAPTER 4. RESULTS
temperature for the star as well as chemical abundances (Shulyak et al., 2004; Khan
& Shulyak, 2006). These effects tend to be small, on the order of 0.1 dex, and hence
would have a minimal impact on our results. However, peculiar He abundance can
have a larger effect. Thus future studies would be well advised to take our results as
a starting point for model atmosphere calculations.
HD 72106A
HD 72106B
v sin i (km s−1 )
ξ (km s−1 )
41.0 ± 0.3
53.9 ± 1.0
2.3 ± 0.6
He
C
O
Mg
Al
Si
Ca
Sc
Ti
Cr
Mn
Fe
Ni
Sr
Y
Ba
Ce
Nd
-2.8 ± 0.3 *
-3.0 ± 0.2 *
-4.0 ± 0.3
-5.8 ± 0.4 *
-3.73 ± 0.14
-5.1 ± 0.5 *
-8.6 ± 0.3 *
-6.04 ± 0.10
-4.33 ± 0.13
-3.49 ± 0.07
-8.7 ± 0.6 *
≤ −8.5 *
-7.5 ± 0.4 *
Solar
-1.07
-3.40 ± 0.08 -3.61
-3.34
-4.60 ± 0.16 -4.47
-5.63
-5.2 ± 0.6
-4.49
-6.0 ± 0.2
-5.69
-9.13 ± 0.08 -8.83
-7.21 ± 0.09 -7.10
-6.3 ± 0.3
-6.36
-6.7 ± 0.5 * -6.61
-4.64 ± 0.17 -4.55
-6.3 ± 0.3
-5.77
-9.2 ± 0.6 * -9.08
-10.0 ± 0.3
-9.79
-10.2 ± 0.5 * -9.83
≤ -9.0 *
-10.30
≤ -9.2 *
-10.54
Table 4.2: Averaged best fit chemical abundances, v sin i and microturbulence (ξ) for
HD 72106A and B as well as solar abundances from Grevesse et al. (2005). Entries
marked by an * are based on only a few lines, and hence have larger ∼ 2σ error bars.
119
CHAPTER 4. RESULTS
3
2
1
0
-1
-2
He
C
O Mg Al
Si Ca Sc
Ti
Cr Mn Fe Ni
Sr
Y
Ba Ce Nd
Figure 4.2: Abundances relative to solar for HD 72106A (black circles) and HD
72106B (red/gray squares), averaged over all spectral windows modeled. The dashed
line at 0 represents solar abundance. Points marked with only an arrow indicate the
value is an upper limit only. Strong departures from solar abundance can be seen for
HD 72106A, whereas HD 72106B has largely solar abundances.
CHAPTER 4. RESULTS
4.1.2
120
Secondary
Initial spectrum modeling suggested that the secondary possessed roughly solar chemical abundances. More detailed modeling showed that our initial temperature and
log g values gave poor fits, and produced abundance values inconsistent with solar,
as described in Sections 3.1.1 and 3.3.2. In particular, multiple lines of a single element often could not be fit well with the same abundance. A closer inspection showed
that lines of lower ionization states tended to be too deep when lines of higher ionization states were well fit; strongly suggesting an incorrect temperature and log g. A
revised effective temperature of 8750 K was adopted with log g = 4.0. No magnetic
field was included in these models, since there is no evidence for the presence of a
magnetic field in this star. Microturbulence (ξ) was included as a free parameter in
the models as well as v sin i. One expects a non-zero microturbulence in HD 72106B,
as microturbulence is usually seen in cool, non-magnetic A type stars (Gray, 2005;
Landstreet, 1998).
As for HD 72106A, large regions of HD 72106B’s spectrum were modeled in several
independent segments, following the procedure outlined in Section 3.3.2. The observation of HD 72106B from March 5, 2007 was used. Best fit results for individual
windows are presented in Table 4.3. Sample best fits of models to the observed spectrum can be seen in Figure 4.3. Abundances, v sin i and Microturbulence averaged
over all modeled windows are presented in Table 4.2 and can be seen graphically in
Figure 4.2.
The large majority of elements are consistent with solar abundances, within 2σ
at most. A few elements appear to depart marginally from solar values, with a
significance slightly greater than 2σ. C appearers to be over-abundant with ∼ 2σ
CHAPTER 4. RESULTS
121
significance, whereas Sc is under-abundant by ∼ 3σ. Nevertheless, nearly all the
elements are within 2σ of solar abundance, and no elements display the strong peculiarities seen in HD 72106A. We find v sin i = 53.9 ± 1.0, which is larger than that
of the primary, but not dramatically so.
One other trend of possible significance is that the best fit abundances are consistently below solar. The abundances are, as mentioned, almost all within uncertainty
of solar, however the distribution of abundances is clearly not Gaussian around the
solar values. This may be indicative of a still incorrect temperature, or an overly
large microturbulence. An increase in temperature to 9000 K increased the best fit
abundances by 0.1 dex or 0.2 dex depending on the element. This temperature results in a marginally worse fit to the observed spectrum, but may be the simplest
explanation for the trend of under-abundance. Alternately, since excellent fits of the
synthetic to the observed spectra are generally seen, the trend may in fact reflect a
systematic lower metallicity in HD 72106B near the surface.
Emission Features in HD 72106B
Emission was observed in the Hα Balmer line and the OI 7773 Å triplet of HD
72106B. A careful examination of the spectrum of the secondary revealed no other
emission lines, at the level of the noise in our spectra. Small amounts of emission
infilling in a small number of lines cannot be completely ruled out, but must not be
much beyond the noise in our observations. Variability between observations in the
emission lines, both in Hα and in the OI 7773 Å triplet, was noted. Observations of
the Hα emission line in the secondary are illustrated in Figure 4.4, and observations
of the OI 7773 Å triplet are illustrated in Figure 4.5.
C
Mg
Si
Ca
Sc
Ti
Cr
Mn
Fe
Ni
Sr
Y
Ba
Ce
Nd
4400-4500 Å
4500-4600 Å
4600-4700 Å
5000-5200 Å
5200-5400 Å
5400-5600 Å
54.2
2.1
52.1
2.5
54.3
3.0
53
1.2
54
2.8
55.2
2.6
54.5
2.0
-3.4
-4.6
-4.7
-3.3
-3.5
-4.4
-5.6
-5.8
-9.2
-7.4
-6.2
-5.7
-9.3
-7.2
-6.5
-4.7
-4.8
-4.6
-6.0
-7.3
-6.7
-4.7
-5.7
-6.3
-7.2
-6.5
-4.9
-9.3
-9.7
-9.1
-7.1
-6.0
-9.0
-7.2
-4.4
-6.0
-4.7
-6.1
-7.3
-6.4
-4.7
CHAPTER 4. RESULTS
v sin i (km s−1 )
ξ (km s−1 )
4170-4265 Å
-4.4
-6.7
-10.3
-10.2
≤ -9.0
≤ -9.2
≤ -8.5
Table 4.3: Chemical abundances for HD 72106B in each independently fit segment of spectrum. Solar abundances
for virtually all elements are found.
122
123
CHAPTER 4. RESULTS
1
Flux
0.9
Fe
Ti
0.8
Ti
Si
Fe
Fe
Ca
Fe
Fe
Ti
Ti
Fe
0.7
Mg
4440
4450
4470
4460
Wavelength (angstroms)
4480
Fe
Cr
Fe
4490
1
Flux
0.95
0.9
0.85
Fe
Sr
Fe
Cr
0.8
4200
4210
Fe
Ca
4220
Fe
Cr
4230
Wavelength (angstroms)
Cr
Fe
Sc
4240
Fe
4250
Figure 4.3: Sample best fit synthetic spectra for HD 72106B in two independently
fit spectral windows. Major contributors to each line have been labeled, in order of
importance. The smooth solid line is the best fit spectrum in this region, the dashed
line is a spectrum computed with solar chemical abundances.
124
CHAPTER 4. RESULTS
1.4
Flux
1.2
1
0.8
0.6
652
654
656
Wavelength (nm)
658
660
Figure 4.4: Emission and variability in the Hα Balmer line of HD 72106B. The black
line represents the individual observation of the secondary from JD 2453748.01496
and the red/gray line is the spectrum of just the secondary from JD 2454164.84650.
125
CHAPTER 4. RESULTS
1.05
1
Flux
0.95
0.9
0.85
0.8
0.75
0.7
776
776.5
777
777.5
Wavelength (nm)
778
778.5
779
Figure 4.5: Emission and variability in the OI 7773 Å triplet of HD 72106B. The black
line represents the individual observation of the secondary from JD 2453748.01496
and the red/gray line is the spectrum of just the secondary from JD 2454164.84650.
CHAPTER 4. RESULTS
126
The emission in Hα is by far the more prominent. The wings of the Hα line are
observed in absorption and the core contains a large single peaked emission feature.
There is a hint of an absorption core superimposed on the emission peak in our
second individual observation of the secondary (at JD 2454164.84650). However, this
absorption feature is very sharp and somewhat off center, thus it may be a telluric
line. Comparing the Hα lines in observations of the combined HD 72106 system
suggests that variability occurs on time scales as short as one day. However, larger,
longer term trends are also present, as the variation between observations separated
by months or years is larger than the variation between days. Comparison of Hα lines
in observations of the combined system, for the purposes of investigating variability
in the secondary, is fairly safe since there is no evidence for variability in the Hα line
of the primary. No clear periodicity is seen in the Hα line variability.
The emission in the OI 7773 Å triplet is more subtle than that of the Hα line.
In our second individual observation of the secondary, a rise above the continuum
level is seen around the edges of the OI 7773 Å triplet. A careful examination of the
unnormalized spectrum shows that this is not a result of the normalization process. It
is unlikely that this is an instrumental artifact, as this rise is observed only in this line
of the secondary, and the feature is too broad to be a result of a bad pixel, a cosmic
ray, or the terrestrial atmosphere. This, combined with the variability in the OI 7773
Å triplet in the observations of the secondary individually, leads us to conclude that
we are observing a small amount of emission in this triplet. The variability in line
depth we attribute to emission infilling of the absorption line. Furthermore, no such
rise above the continuum level is observed in our spectra of the primary on its own,
and while there may be some variability in the OI 7773 Å triplet of the primary,
CHAPTER 4. RESULTS
127
it is of much smaller amplitude than that observed in the secondary. This provides
further evidence that the observed feature is indeed emission, and that it is intrinsic
to HD 72106B.
The observed emission in Hα and the OI 7773 Å triplet of HD 72106B, combined
with the star’s H-R diagram position, fully support this star’s classification as a
HAeBe star. This combined with the star’s infrared excess (Oudmaijer et al., 1992),
and the presence of molecular bands in the star’s infrared spectrum (Schütz et al.,
2005), place HD 72106B firmly in the HAeBe class.
4.2
Rotation Period of the Primary
The rotation period of HD 72106A was determined using the procedure described in
Section 3.6. Mean periodograms based on the time series of pixels comprising the
Stokes I and V profiles, illustrated in Figure 3.10, display a few minima with reduced
χ2 values of about 2. Following the method described by Press et al. (1992) and
Avni (1976) (discussed in Section 3.6), confidence limits of constant χ2 were placed
on the periodogram. The χ2 minima at 0.38983 days, 0.63995 days, 1.6921 days and
1.7859 days were all within the 99% confidence limit, and hence were considered to
be possible rotation periods.
The phase variation of the Stokes I and V LSD profiles for each candidate period
was examined visually. The data were phased according to each rotation period, and
the LSD profiles plotted according to phase. Phased LSD profiles were then examined to verify that the resultant variation was qualitatively consistent with rotational
modulation. The 0.63995 day period produced the best phasing of the profiles, as
128
CHAPTER 4. RESULTS
400
Bz (G)
300
200
100
0
-0.2
0
0.2
0.4
Phase
0.6
0.8
1
1.2
Figure 4.6: Longitudinal magnetic field measurements of HD 72106A, phased with
the adopted 0.63995 day period, and the best fit sinusoid. A good fit of the sinusoid
to the data is seen with a reduced χ2 of 0.86.
illustrated in Figure 4.7. This period also provided a good phasing of the longitudinal magnetic field data, illustrated in Figure 4.6. The 1.6921 and 1.7859 day periods
produced less satisfactory variations. Each of these periods resulted in small phase
differences between observations exhibiting large profile differences. These variations
were considered unphysical and the associated periods were discarded. Figure 4.8
illustrates the phasing of profiles for the 1.7859 day period and highlights LSD profiles in an unphysical order, Figure 4.9 does the same for the 1.6921 day period. The
0.38983 day period produced also unphysical variations at a few phases, as illustrated
in Figure 4.10. Additionally, the 0.38983 day period would require an inclination
i ∼ 0◦ to be compatible with the observed v sin i. Finally, this period is remarkably
short - smaller than that of any known Ap star. The shortest known rotation periods
of Ap stars are around 0.5-0.6 days (Adelman et al., 1992; North, 1998; Kochukhov
& Bagnulo, 2006; Power, 2007).
Thus we conclude that the 0.63995 day minimum represents the most probable
129
CHAPTER 4. RESULTS
rotation period of HD 72106A. Using the 99% confidence limits described above, we
obtain the rotation period of 0.63995 ± 0.00014 days (3σ). This is notable in that it
is one of the shortest rotation periods seen in any Ap/Bp star.
Wade et al. (2005) reported a possible rotation period near 2 days. This period
can be confidently ruled out by both the periodograms and the phasing of LSD profiles it produces. However, given the authors’ limited data set, it is not surprising
that they found this period. The star, with our 0.63995 ± 0.00014 day period, rotates
approximately three times in two days, and ∼1.5 times in one day. So observations
on consecutive nights reveal approximately diametrically opposite phases and observations spaced two nights apart reveal the same rotation phase. Thus Wade et al.
(2005) observed an alias of the true period. Our data set remedies this problem by
having observations that are more closely spaced in time than one day.
With an accurate rotation period, it is now possible to calculate the inclination
angle (i) of the star’s rotation axis to our line of sight. Given the radius of 1.3 ± 0.6 R ,
as determined in Section 3.1.2, one can calculate the circumference of the star. Using
the rotation period one can then easily calculate the rotation velocity at the equator
of the star (v). Then using the definition of v sin i and the value v sin i = 41.0 ± 0.7
km s−1 from Section 4.1.1, one can calculate i. That is:
sin i =
P v sin i
,
2πR
(4.1)
where P is the rotation period and R is the stellar radius. Performing this calculation
produces the value i = 23 ± 11◦ . Knowledge of the inclination angle is essential for
accurate modeling of the magnetic field geometry.
130
CHAPTER 4. RESULTS
0.00500
1
0.00500
1
0.00800
0.00800
0.10638
0.95
0.10638
0.12878
0.995
0.12878
0.15286
0.15286
0.17770
0.9
0.17770
0.99
0.22480
0.22480
0.25827
0.85
0.25827
0.30952
0.985
0.30952
V/I
I
0.32974
0.32974
0.33870
0.8
0.33976
0.33870
0.98
0.33976
0.51048
0.75
0.51048
0.53004
0.76104
0.975
0.53004
0.76104
0.80075
0.7
0.89434
0.80075
0.97
0.96778
0.89434
0.65
0.96778
-50
0
50
Velocity (km/s)
100
0.965
-50
0
50
100
Velocity (km/s)
Figure 4.7: Phased LSD profiles (Stokes I on the left and Stokes V on the right) for
HD 72106A with the adopted 0.63995 day best fit period. The profiles are labeled
according to phase, and have been shifted vertically for display purposes.
131
CHAPTER 4. RESULTS
0.01098
1
0.01098
1
0.01732
0.01732
0.10264
0.95
0.10264
0.22180
0.995
0.22180
0.25067
0.25067
0.25456
0.9
0.25456
0.99
0.27121
0.27121
0.29615
0.85
0.29615
0.32735
0.985
0.32735
V/I
I
0.33697
0.33697
0.40960
0.8
0.41235
0.40960
0.98
0.41235
0.47353
0.75
0.47353
0.56380
0.84664
0.975
0.56380
0.84664
0.86171
0.7
0.90267
0.86171
0.97
0.99657
0.90267
0.65
0.99657
-50
0
50
Velocity (km/s)
100
0.965
-50
0
50
Velocity (km/s)
100
Figure 4.8: Phased LSD profiles (Stokes I on the left and Stokes V on the right) for
HD 72106A with the incorrect 1.7859 day period. The profiles are labeled according to
phase, and have been shifted vertically for display purposes. Note the inconsistencies
between phases 0.22180 and 0.27121 (profiles in bold). Such large changes, particularly noticeable in the set of Stokes I profiles, over only 5% of the star’s rotation are
highly unlikely.
132
CHAPTER 4. RESULTS
0.01159
1
0.01159
1
0.04671
0.04671
0.10319
0.95
0.10319
0.15780
0.995
0.15780
0.17538
0.17538
0.23463
0.9
0.23463
0.99
0.28338
0.28338
0.32645
0.85
0.32645
0.34496
0.985
0.34496
V/I
I
0.37543
0.37543
0.37687
0.8
0.46145
0.37687
0.98
0.46145
0.52602
0.75
0.52602
0.62348
0.79861
0.975
0.62348
0.79861
0.86438
0.7
0.87026
0.86438
0.97
0.99638
0.87026
0.65
0.99638
-50
0
50
Velocity (km/s)
100
0.965
-50
0
50
Velocity (km/s)
100
Figure 4.9: Phased LSD profiles for HD 72106A with the incorrect 1.69219 day period,
as in Figure 4.8. Note the inconsistencies at phases 0.23465, 0.32845 and 0.34496.
133
CHAPTER 4. RESULTS
0.00375
1
0.00375
1
0.05030
0.05030
0.07208
0.95
0.07208
0.08003
0.995
0.08003
0.17857
0.17857
0.23033
0.9
0.23033
0.99
0.26070
0.26070
0.29526
0.85
0.29526
0.31083
0.985
0.31083
V/I
I
0.33722
0.33722
0.34389
0.8
0.41731
0.34389
0.98
0.41731
0.57561
0.75
0.57561
0.62414
0.75225
0.62414
0.975
0.75225
0.78524
0.7
0.96557
0.78524
0.97
0.98428
0.96557
0.65
0.98428
-50
0
50
Velocity (km/s)
100
0.965
-50
0
50
Velocity (km/s)
100
Figure 4.10: Phased LSD profiles for HD 72106A with the incorrect 0.38983 day
period, as in Figure 4.8. Note the inconsistencies between phases 1.7857 and 0.28070.
CHAPTER 4. RESULTS
4.3
134
Magnetic Field
The phase variation of the longitudinal field of HD 72106A exhibits an approximately
sinusoidal form. This implies that the underlying field has an important large scale
dipole component. Therefore we adopt an oblique dipole rotator model, as discussed
in Section 3.5, to describe the magnetic field of HD 72106A. A first-order sinusoid
was fit by least squares through the longitudinal field data, as discussed in Section
4.2. The 0.63995 day rotation period was used, with the phase, amplitude and mean
value of the sinusoid as the free parameters in the fit. This produced a good fit
(reduced χ2 = 0.86) which is shown in Figure 4.6. According to this fit, the maximum
longitudinal field is +360 ± 40 G while the minimum is +65 ± 40 G. Computing the
r parameter, described in Section 3.5, we obtain r = 0.18 ± 0.11. From equation 3.20
with the inclination angle, discussed in Section 4.2, i = 23 ± 11◦ we find β = 58 ± 15◦ .
Then from equation 3.19 we compute the polar dipole field strength, Bp = 1490 ± 360
G. A standard limb darkening coefficient for a late B star, u = 0.4, was used (Gray,
2005). These values are all given with 1σ error bars.
This dipole field model was verified by using ZEEMAN2 to model the full set of
phased Stokes V LSD profiles, as described in Section 3.5. First, a dipole field with
the values i = 23◦ , β = 58◦ , and Bp = 1490 G were used for the input geometry.
Profiles were synthesized with ZEEMAN2 for all observed LSD profiles of the primary,
resulting in 18 different phases. The models were compared by eye and by χ2 to the
Stokes V LSD profiles. No major discrepancies were observed in either the shapes
or amplitudes of the Stokes V profiles, and a reduced χ2 of 2.87 was calculated.
From this we concluded that the dipole field model derived from the longitudinal
field measurements is able to fit the Stokes V observations to nearly within the error
CHAPTER 4. RESULTS
135
bars. It is not surprising that the fit is less than perfect, because the Stokes I profiles
were fit only approximately.
Additionally, a grid of dipole field models was calculated and compared to the
observed Stokes V LSD profiles. A coarse grid using 100 G steps in the dipole strength
and 10◦ steps in β was first calculated to search from 0 to 2500 G and 0◦ to 180◦ . A
finer grid, using 20 G steps in Bp and 1◦ steps in β was calculated, centered on the χ2
minimum from the coarse grid. A map of χ2 for the models tested is shown in Figure
4.11. A minimum was found at Bp = 1300 G and β = 60◦ , with a reduced χ2 of 2.6184
(for 522 data points: 29 points per profile at 18 different phases). A comparison of
the best fit model Stokes V profiles to the observed profiles can be seen in Figure 4.12.
Uncertainties were determined following the method described by Press et al. (1992),
which was also used for the period determination in Section 4.2. To take into include
the large uncertainty on the inclination angle (i = 23 ± 11◦ ), a grid of models was
calculated at the maximum and minimum allowed i. The χ2 minimum was found,
and confidence ellipses at the 3σ level were calculated. The final uncertainty is then
the total range of values allowed, at the 3σ limit, at the extrema in i. From this we
derive the values Bp = 1300 ± 100 G, and β = 60 ± 5◦ . However, as described in
Section 3.5 the shapes of the Stokes I profiles were not modeled, hence our confidence
in the details of the Stokes V profile models is decreased. Consequently, the formal
uncertainties presented here may be somewhat too small. However, the derived field
parameters are in good agreement with those derived from the longitudinal magnetic
field measurements, therefore the true magnetic field geometry cannot depart radically
from the one presented here.
In HD 72106B we detect no magnetic field. Our two observations of the secondary
136
CHAPTER 4. RESULTS
i = 12o
80
2
χ
β (o)
75
2.455
70
2.45
65
2.445
2.44
60
2.435
55
50
45
40
1000
1100
1200
Bp (G)
1300
1400
1500
i = 23o
80
2
χ
75
2.64
70
2.635
2.63
β (o)
65
2.625
60
2.62
55
50
45
40
1000
1100
1200
Bp (G)
1300
1400
1500
i = 34o
80
χ2
75
2.97
70
2.965
β (o)
65
2.96
60
2.955
55
50
45
40
1000
1100
1200
Bp (G)
1300
1400
1500
Figure 4.11: Maps of reduced χ2 for a range of dipole field models for HD 72106A at
different inclination angles. The minimum in χ2 can be seen in the i = 23◦ map at
Bp = 1300 G and β = 60◦ . The reduced χ2 scales correspond to the 3σ (∼ 99.75%)
confidence limit, calculated using the probability tables of Press et al. (1992).
137
CHAPTER 4. RESULTS
0.00500
1
0.00800
0.10638
0.995
0.12878
0.15286
0.17770
0.99
0.22480
0.25827
0.985
0.30952
V
0.32974
0.33870
0.98
0.33976
0.51048
0.975
0.53004
0.76104
0.80075
0.97
0.89434
0.96778
0.965
5232.5
5233
5233.5 5234 5234.5
Mean Wavelength (A)
5235
5235.5
Figure 4.12: Synthetic Stokes V LSD profiles (lines) corresponding to the best fit
magnetic field geometry (i = 23◦ , Bp = 1300 G, β = 60◦ ) compared with observed
profiles (points). The profiles are labeled by phase, and acceptable fits are obtained
at all phases.
CHAPTER 4. RESULTS
138
individually yield longitudinal magnetic field measurements of 0 ± 170 G and -50 ±
60 G. Thus we can place an upper (1σ) limit on the longitudinal field in the secondary
at 170 G (since the star may have been at a less favorable rotation phase when the
more accurate measurement was made). Additionally, the Stokes V LSD profiles for
both observations are consistent with the null field hypothesis. This rules out the
possibility of a large scale field being present but undetected due to the rotation
phase of the star (such as a dipole aligned perpendicular to the line of sight). If such
a magnetic field were present, one would still see a ‘cross-over signature’ in the Stokes
V profile.
In their FORS1 observation of HD 72106B Wade et al. (2005) find no magnetic
field detection, reporting a longitudinal field measurement of 60 ± 55 G. Wade et
al. (2007) re-analyzed the same observation, in a more robust fashion, and report
longitudinal fields of 52 ± 90 G from an analysis of Balmer lines and 3 ± 122 G
from an analysis of metallic lines. These results provide further evidence that the
secondary is non-magnetic, further decreasing the chance that it was simply observed
at an inopportune rotation phase. Thus we conclude that if the secondary has a
magnetic field, either it is not globally ordered, or it has a longitudinal field strength
that never exceeds ∼ 200 G.
4.4
Surface Abundance Geometry
Motivated by the asymmetries and phase variability seen in many metallic line profiles, we undertook a detailed analysis of the surface abundance distribution of several
elements in HD 72106A. Doppler Imaging with the INVERS12 code was used to perform this analysis, as described in Section 3.7. For these models a 30◦ inclination
CHAPTER 4. RESULTS
139
angle was assumed, the adopted 0.63995 day rotation period was used, and Teff =
11000 K, and log g = 4.0 were assumed for the model atmosphere. This inclination
angle is not quite our best fit value, but it is well within uncertainty, and the difference would have a minimal impact on our results. The initial value for v sin i was 41
km s−1 , and abundances for the treatment of blends were taken from Table 4.2.
Initially the Doppler Imaging procedure was performed using five Si II lines: 4128
Å, 4130 Å, 5056 Å, 5041 Å, and 6371 Å. These lines were strong enough to have
sufficient S/N for Doppler Imaging, and also possessed strong variability. The five
lines were modeled simultaneously. The reconstructed surface abundance distribution
for Si is illustrated in Figure 4.13. The best fit models of the Si lines used are presented
in Figure 4.14. A clear spot of over-abundance can be seen in the northern hemisphere,
with a ring of Si over-abundance near the equator. The Si depletion visible in the
southern hemisphere is likely an artifact of the poor sensitivity of the procedure to
this mostly hidden region of the star.
Figure 4.13: Surface abundance map of Si for HD 72106A. This map is based on five
Si II lines: 4128 Å, 4130 Å, 5056 Å, 5041 Å, and 6371 Å. A good agreement of the
large scale abundance distribution is obtained with the Si map reconstructed from
the LSD profiles (see Figure 4.15), although the scale of the abundances does not
match (for reasons described in the text). The ‘X’ represents the rotational pole, the
green circle indicates the rotational equator. The abundance scale on the right is in
Si
.
units of log NNtot
CHAPTER 4. RESULTS
140
Figure 4.14: Line fits for individual Si II lines used in Doppler mapping of HD 72106A.
Observed line profiles are points and synthetic profiles are solid lines. Contributors
to each line have been labeled. The bars in the lower left indicate the vertical and
horizontal scale, 2.5% of the continuum and 0.5 Å respectively. Fairly good fits to
the line profiles can be seen. However, there is a large amount of noise relative to the
observed variability.
CHAPTER 4. RESULTS
141
Problematically, there were very few strong lines with both very high S/N and
large variability in our spectra of HD 72106A. Stronger, more saturated lines inherently display less variability than their weaker, less saturated counterparts. This,
combined with the relatively low S/N ratio of the observed spectra, resulted in very
few lines being suitable for Doppler Imaging. The limited suitability of our spectra
for Doppler Imaging is largely due to the use of Doppler Imaging not being foreseen when many of the spectra were originally collected. HD 72106A, at almost 9 th
magnitude, is a relatively faint target for Doppler Imaging. Originally, shorter exposure times, allowing for better phase coverage, were chosen. Our ability to perform
Doppler Imaging at all is largely a ‘bonus’, thanks to the large number of spectra
collected for rotation period determination and magnetic field modeling.
Doppler Imaging using LSD profiles, as described in Section 3.7, was undertaken to
remedy the problem of insufficient S/N. Suitable LSD profiles, constructed with line
masks containing lines of individual elements, were generated for Si, Ti, Cr, and Fe.
These LSD profile time series were then used as the input data for Doppler Imaging.
Adopted atomic data were constructed from the average properties of the lines used in
LSD for each element. The depth of the LSD profile is a complex function of the lines
used in the analysis. It is therefore not possible to assign a realistic oscillator strength
to the LSD profiles. Consequently a log gf of 1 was adopted for all elements, and
abundances were scaled to fit the LSD profile. Therefore the maps reconstructed by
Doppler Imaging from LSD profiles should reflect the correct distribution of surface
features (patches of over- or under-abundance), and probably the correct abundance
contrast, but the absolute abundance scale is unknown.
Surface abundance maps inferred from Doppler Imaging of LSD profiles for Si, Ti,
CHAPTER 4. RESULTS
142
Cr, and Fe are illustrated in Figure 4.15. Fits of the synthetic to the observed LSD
Stokes I profiles are shown in Figure 4.16. Several interesting features are evident from
the Doppler Imaging maps. Ti, Cr, and Fe all seem to share very similar abundance
patterns in the northern hemisphere. This is reflected in the similar phase variations
of the LSD profiles in Figure 4.16. A large patch of over-abundance centered near
phase 0 is apparent in all three maps, with another somewhat smaller over-abundance
spot about 180◦ away in longitude, at the same latitude, around phase 0.6. The Si
map shares the larger spot but not the smaller, as reflected in its LSD profiles. The
large over-abundance ring at the equator in the Si map, and set of spots around the
equator in the Ti map, are probably real. The ring is not evident in the Cr and Fe
maps, but one is apparent in the Si map based on several lines in the original spectra.
However, such equatorial features are vulnerable to systematic effects, as are features
centered at the rotational pole. There appears to be a large over-abundance spot in
the southern hemisphere for both Fe and Cr, although the sensitivity of the map is
poor in that location. When the magnetic field geometry, determined in Section 4.3,
is compared to the Doppler maps, it appears that the positive magnetic pole lies near
the large spot of over-abundance at phase 0 in all four maps. However, the magnetic
pole lies roughly to the south and west of the center of the abundance spot, thus
the relationship is not entirely clear. The over-abundance spots cannot be aligned
with the rotational pole, as they would not generate the observed variability in that
position..
Substantial abundance contrasts are observed in the Doppler Imaging maps. Ti
shows a 2 dex contrast. Fe shows a 2.5 dex contrast, if the spots of strong overand under-abundance in the southern hemisphere are to be believed, and a 1.5 dex
CHAPTER 4. RESULTS
143
contrast in the northern hemisphere. Cr displays a 4 dex contrast overall, but only a
2 dex contrast in the more reliable northern hemisphere. Similarly, Si displays a 4.5
dex contrast overall, with a contrast of 1.5 dex in the northern hemisphere
It is useful to consider HD 72106A in the context of other Ap/Bp stars for which
multi-element Doppler Imaging has been performed. The elements He, Mg, Si, Cr,
and Fe were mapped in the Bp star CU Virginis (HD 124224) with Doppler Imaging
by Kuschnig et al. (1999). They found a temperature of 13000 K, a log g of 4.0 and
quote a rotation period of 0.5207038 days, making CU Virginis somewhat similar to
HD 72106A. Trigilio et al. (2000) found a dipole magnetic field strength of 3000 ± 200
G and an obliquity of 74 ± 3◦ . The Doppler maps of Si, Cr, and Fe for CU Virginis
were all very similar to one another. For all three elements, a large region of depletion,
relative to the global average, was seen near the magnetic pole, at ∼60◦ latitude. A
large spot of over-abundance was seen for all three elements at the same latitude,
but 180◦ away in longitude. Helium had an opposite pattern to that observed in Si,
Cr, and Fe, with a large over-abundance spot near the magnetic pole and a large
depletion 180◦ away. In light of this, it is perhaps not surprising that Si, Cr, and Fe
display similar surface abundance distributions in HD 72106A.
Lueftinger et al. (2003) found somewhat similar results for Ursae Majoris. Ursae Majoris (HD 112185) is a 9000 K (log g = 3.6) Ap star with ∼ 5 day period and
a dipole field strength of several hundred gauss. Lueftinger et al. (2003) constructed
surface maps of Ti, Cr and Fe, among other elements, with Doppler Imaging. They
found distributions of Cr and Fe very similar to each other, with two large spots
of over-abundance near the longitude of the magnetic poles. Ti was roughly anticorrelated with Fe and Cr, displaying two large spots of under-abundance at the
CHAPTER 4. RESULTS
144
same positions as the over-abundance spots of Cr and Fe.
In the cooler roAp star HR 3831 (HD 83368) Kochukhov et al. (2004) found
somewhat different results. HR 3831 is a rapidly oscillating Ap star (roAp), with a
∼ 2.9 day rotation period, a 11.67 min pulsation period and a temperature of 7700 K.
As such, it may provide a poor comparison to HD 72106A. However, a brief mention
of this star is worthwhile, if only to illustrate the differences observed between Ap/Bp
stars. In this star Kochukhov et al. (2004) found no clear correlation between the
distribution of Si, Ti, Cr, or Fe. The authors found over-abundance features near the
magnetic poles for many elements, but the details varied widely.
In this context, the surface abundance distribution patterns seen in HD 72106A
are roughly consistent with those seen in similar Bp stars. However, the details are
particular to HD 72106A itself. For example, over-abundances of Fe and Cr are seen
near the north magnetic pole, as in Ursae Majoris, however the pattern of Ti follows
that of Cr and Fe, unlike in the case of Ursae Majoris. It remains unclear whether
these discrepancies represent the range of variation intrinsic to Ap/Bp stars, or if the
are a result of HD 72106A’s young evolutionary status.
CHAPTER 4. RESULTS
145
Figure 4.15: Surface abundance maps of HD 72106A for Si, Ti, Cr and Fe. The maps
are all based on fits to LSD profiles. The ‘X’ represents the rotational pole, the green
circle indicates the rotational equator. The abundance scale on the right is in units
Si
.
of log NNtot
CHAPTER 4. RESULTS
146
Figure 4.16: Fits of synthetic Stokes I LSD profiles to observed LSD profiles for Si,
Ti, Cr, and Fe. The profiles are labeled according to element and phase. The bars
in the lower left indicate the vertical and horizontal scale, 5% of the continuum and
0.5 Å respectively. Generally good fits can be seen, ranging from the high S/N Fe
profiles at the best to the noisy Si profiles at the worst.
Chapter 5
Summary, Discussion and
Conclusions
5.1
Summary of Results
In this thesis 22 high resolution spectropolarimetric observations of the HD 72106
system were analyzed in detail. This system is composed of a Herbig Ae secondary
and a magnetic, chemically peculiar primary. The observations were obtained with the
ESPaDOnS instrument at the CFHT. Spectra were recorded in circular polarization
as well as total intensity.
Based on Balmer line fitting we determined Teff = 11000 ± 1000 K and log g =
4.0 ± 0.5 for the primary. For the secondary, with the additional aid of spectrum
synthesis, we determined Teff = 8500 ± 500 K and log g = 4.0 ± 0.5. Using Hipparcos
measurements and placing the stars on the H-R diagram, we found a mass of 2.4 ±
0.4 M and a radius of 1.3 ± 0.6 R for the primary. For the secondary we found
M = 1.8 ± 0.2 M and R = 1.4 ± 0.6 R . If one calculates log g from mass and
147
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
148
radius one finds values consistent with those presented above, with comparable error
bars, thus we prefer the more directly inferred log g values.
We find that HD 72106B is conclusively a Herbig Ae star. The emission in both
the Hα Balmer line and the OI 7773 Å triplet, together with the star’s H-R diagram
position strongly support the HAeBe nature of the star. In addition HD 72106B
displays an infrared excess (Oudmaijer et al., 1992), and has an infrared spectrum
containing a number of molecular lines (Schütz et al., 2005). Thus there is little doubt
of HD 72106B’s true HAeBe nature.
The age of the HD 72106 system, a critical parameter for this analysis, proved to
be somewhat uncertain. We find the range 3 Myr to 13 Myr consistent with the H-R
diagram positions of the stars, with a best fit value of 10 Myr. Taking the younger
end of this range implies that the entire system is on the pre-main sequence, and that
the primary has about 1 Myr left before it reaches the main sequence. In the older
limit, the secondary is nearing the ZAMS and the primary has been on the main
sequence for about 9 Myr, giving it a fractional main sequence age (τ ) of 0.015. In
the most likely case (10 Myr) the primary reached the main sequence about 6 Myr
ago and has a fractional age τ of 0.01. Thus we cannot confidently conclude that HD
72106A is a pre-main sequence object, but even if it is on the main sequence it is still
very young.
Surprisingly strong over-abundances, relative to solar abundances, of a number
of chemical elements were found in the photosphere of HD 72106A. Fe was found
to be over-abundant by almost 10 times, as was Si. Cr was over-abundant by 100
times and Nd was over-abundant by 1000 times. Conversely, He was found to be
under-abundant, well below 0.1 times solar. A number of other elements, such as
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
149
Al and Sc, have solar abundances within their uncertainties. These peculiarities are
characteristic of Ap and Bp stars, although the peculiarities seen in HD 72106A would
be fairly strong by Ap/Bp standards. This makes their appearance in this very young
star that much more surprising.
HD 72106B, unlike HD 72106A, shows abundances that are generally within uncertainty of solar abundances. A few marginal departures may exist, particularly for
C and Sc. On average, the abundances derived are slightly below solar. This may
indicate a truly lower metallicity in the star, not inconsistent with that observed in
normal A stars, or it may be only an artifact due to the uncertainty in the temperature of this star. The existence of this chemically normal A-type star in a binary
system with a strongly chemically peculiar primary adds to the list of interesting
properties of HD 72106.
The rotation period of HD 72106A was carefully examined. Based on longitudinal
field measurements, as well as variability in the set of Stokes I and V LSD profiles, we
determined the rotation period to be 0.63995 ± 0.00005 days. This value is notable
in that it is a particularly short rotation period for an Ap/Bp star. The most rapidly
rotating Ap star in our vicinity (within 100 pc of the Sun) is HD 124224 (Power, 2007),
rotating once in 0.52068 days (Adelman et al., 1992). The majority of Ap/Bp stars
have periods in the range 1 to 10 days, although much longer periods are observed in
some cases (Kochukhov & Bagnulo, 2006).
The magnetic field geometry of HD 72106A was determined by fitting both the
longitudinal field curve and the set of phased Stokes V profiles. Based on the derived
rotation period, radius, and v sin i, the inclination angle of the star is i = 23 ± 11◦ .
We then find that the dipole magnetic field has an obliquity of β = 60 ± 5◦ and a
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
150
polar field strength of Bp = 1300 ± 100 G. A simple centered oblique dipole magnetic
field reproduces all observations very well. Thus we conclude that, if higher order
components to the field exist, they are much weaker than the dipole component. We
also conclude that the magnetic field is stable on time scales of several years. The
fact that we find a good phasing for the longitudinal field measurements and Stokes V
profiles with a single period, despite observations being separated by up to two years,
suggest that the magnetic field cannot have changed much over that time period.
Doppler Imaging of surface abundance inhomogeneities was performed for HD
72106A. While Doppler Imaging using individual lines was attempted, superior results
were achieved by using LSD profiles tailored to specific elements. Surface abundance
maps were made for Si, Ti, Cr, and Fe, all of which showed clear inhomogeneities.
Patches of overabundance corresponding roughly to the positive magnetic pole were
seen for all four elements. As well, patches or rings around the rotational equator
were seen for a few elements, and were particularly clear for Si. However, a number of
surface abundance features with no obvious correlation to magnetic field or rotation
were also seen. These maps clearly show that HD 72106A possesses surface abundance
inhomogeneities similar to Ap/Bp stars, in addition to peculiar abundances and a
dipolar magnetic field.
5.2
Discussion and Conclusions
These results show that HD 72106A is one of the youngest intermediate mass stars
observed to have a magnetic field. Even if the star has reached the pre-main sequence, it has a remarkably small fractional age. The oblique dipolar geometry of the
magnetic field seen in HD 72106A has only been conclusively shown in one younger
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
151
star, HD 200775A (Alecian et al, 2006; Alecian, 2007). HD 72106A is most likely the
youngest star in which strong chemical peculiarities have been observed. Certainly it
has a fractional age, even with the large uncertainties, that can only be approached
by very few known Ap stars (Bagnulo et al., 2003, 2004; Landstreet et al., 2007).
Of these stars with rival fractional ages, NGC 2244-334 is the only one for which a
detailed abundance analysis has been performed (Bagnulo et al., 2004). Additionally, HD 72106A is the youngest star for which surface abundances inhomogeneities
have unequivocally been shown. The Doppler reconstructions presented here are the
earliest stage of intermediate mass stellar chemical evolution ever mapped.
The set of properties observed in HD 72106A is perfectly characteristic of a main
sequence Ap/Bp star. Thus we conclude that HD 72106A is, for all intents and
purposes, a Bp star. If it is still on the pre-main sequence one could debate the
semantics, however all observations indicate that the photosphere of HD 72106A
is identical to that of a Bp star. This makes HD 72106A possibly the youngest
known Ap/Bp star; certainly it has one of the smallest fractional ages (Bagnulo et
al., 2004; Landstreet et al., 2007). The classification of HD 72106A as a Bp star lends
strong support to the identification of magnetic HAeBe stars as the precursors to
main sequence Ap/Bp stars. The existence of this star suggests that there may be a
continuum of Ap/Bp-like stars back into the late pre-main sequence.
For comparison, consider the Bp star NGC 2244-334. This star has been on the
main sequence for only 2.3 ± 0.3 Myr (Bagnulo et al., 2004; Hensberge et al., 2000),
implying a fractional main sequence age τ = 0.02 ± 0.01. Thus HD 72106A is most
likely the younger star, or at least has a younger fractional age, however that is not
entirely certain. Bagnulo et al. (2004) found that NGC 2244-334 has a mass of 4.0±0.5
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
152
M , a temperature of 15000 ± 1000 K, and observed a longitudinal magnetic field of
9000 G, implying a dipole field strength significantly of at least 30 kG. Bagnulo et al.
(2004) derive a strong under-abundance of He (−2.45 ± 0.3 dex, 1.38 dex below solar)
and remarkably strong over-abundances of Si (−3.5 ± 0.3 dex, 1.0 dex above solar),
Ti (−5.1 ± 0.3 dex, 2.0 dex above solar), Cr (−3.9 ± 0.3 dex, 2.5 dex above solar), and
Fe (−3.3 ± 0.2 dex, 1.3 dex above solar). These abundances are similar to those of
HD 72106A: Ti is even more abundant in NGC 2244-334, however abundances for the
other elements are roughly within uncertainty of each other. This suggests that there
are important similarities between these two very young stars, despite their difference
in mass and large difference in magnetic field strength.
Several pre-main sequence stars also provide useful comparisons. The star HD
200775A is the pre-main sequence magnetic primary of a double-lined binary system
(Alecian et al, 2006; Alecian, 2007). It has a temperature of 18600±2000 K, a mass of
10 ± 2 M and an age of about 0.1 Myr (Hernández et al., 2004; Alecian et al., 2007,
in preparation). Alecian et al. (2007, in preparation) find that a dipole magnetic field
of β = 90◦ and Bp = 400 G, coupled with a 4.3 day rotation period and i = 17◦ ,
reproduce their 21 observations well. The star V380 Ori is a 2.8 ± 0.3 M star with
a temperature of about 10700 K and an age of about 1 Myr as measured from the
birth line (Wade et al., 2005). Wade et al. (2005) find a longitudinal magnetic field of
460 ± 70 G, and Alecian et al (2006) find that the magnetic field has a major dipole
component, but were unable to derive an accurate geometry from their observations.
Wade et al. (2005) and Wade et al. (2007) detect a magnetic field in the star HD
101412, with a longitudinal strength of 512 ± 111 G. Wade et al. (2007) report a mass
of 2.6 ± 0.3 M and and age of about 2 Myr for HD 101412, however they could not
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
153
constrain the magnetic field geometry. A magnetic field was reported in the star HD
104237 by Donati et al. (1997), and confirmed by Donati (2000), with a longitudinal
strength of ∼50 G. This star has a mass of about 2.3 M and an age of about about
2 Myr (van den Ancker et al., 1998). Acke & Waelkens (2004) studied chemical
abundances in HD 104237, using equivalent widths. They find approximately solar
abundances for a range of elements, including Si, Cr, and Fe. The star HD 190073 was
reported to possess a magnetic field by Catala et al. (2007). They find a longitudinal
magnetic field strength of 74 ± 10 G, with no variability in either the longitudinal
field measurements or the Stokes V profiles in 13 months of observation. Catala et al.
(2007) derive a mass of 2.85 ± 0.25 M and an age of 1.2 ± 0.6 Myr (measured from
the birth line). Acke & Waelkens (2004) also studied chemical abundances in this star
using equivalent widths, and found roughly solar abundances. Thus we see magnetic
field strengths (and in the case of HD 200775A a magnetic field geometry) in most
other magnetic HAeBe stars that are similar to that observed in HD 72106A. However,
there may be significant differences in surface chemistry between HD 72106A many
other magnetic HAeBe stars.
Another important comparison to make is between HD 72106A and HD 72106B.
The stars have similar masses (2.4 ± 0.4 M and 1.8 ± 0.2 M respectively) and temperatures (11000 ± 1000 K and 8750 ± 500 K respectively), and identical ages. The
stars also have very similar v sin i values (41.0 ± 0.3 km s−1 and 53.9 ± 1.0 km s−1
respectively). The existence of HD 72106A and B in such a young a binary system
suggests that the two stars formed together. This implies that the initial gas both
stars formed out of was very similar (e.g. Carrier et al., 2002). It is improbable that
there were any great contrasts in temperature, density, abundance, or magnetic field
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
154
between the two stars initially. This makes it all the more surprising that the primary displays a strong dipolar magnetic field and very strong chemical peculiarities,
while the secondary has no observable magnetic field, at the level of the noise in our
observations, and it displays solar chemical abundances. Such results have been seen
in other Ap/Bp stars for both binary systems and clusters (e.g. Carrier et al., 2002;
Silvester, 2007). However, HD 72106 is one of, youngest case of this phenomenon
known. The system HD 200775 the only other intermediate mass pre-main sequence
binary known to possess a star with a strong globally ordered magnetic field (the
primary in that case). Interestingly, the stars in this system are also fairly similar to
each other in both mass and age. It remains an open question why the stars of HD
72106 are so similar in some respects and so different in others.
These observational results make some suggestions about the underlying physical
processes at play. The conclusion that strong, globally ordered magnetic fields exist
on the pre-main sequence for some HAeBe stars is supported. This conclusion is
perfectly consistent with the fossil field theory for the origin of the magnetic field.
The contemporaneous dynamo theory, on the other hand, is presented with further
difficulties in timescale. A strong magnetic field must be generated, propagate to the
surface, and become predominantly an oblique dipole, all before the star reaches the
main sequence. While this is not conclusive evidence against contemporaneous dipole
models, it does provide strong constraints on any such models.
The presence of chemical peculiarities in HD 72106A indicates that atomic diffusion can generate peculiarities on very short time scales, relative to the stellar
life time. This is not especially surprising given the theoretical discussion presented
in Section 1.2 (e.g. Michaud, 1970), however this is some of the first observational
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
155
evidence to constrain the timescale. HD 72106A raises the possibility that atomic
diffusion can generate peculiarities on the pre-main sequence; that is, on time scales
substantially shorter than the pre-main sequence lifetime of the star. Additionally,
these results show that inhomogeneities can develop on such short timescales in the
horizontal direction as well as the vertical. This suggests the effects of rotation and
magnetic fields on diffusion also operate very quickly.
The possibility of diffusion generated peculiarities on the pre-main sequence raises
questions about the impact of accretion. A heavily accreting star almost certainly
does not possess a sufficiently stable atmosphere for chemical peculiarities to build
up via atomic diffusion (e.g. Vauclair, 1981). However, in a star that is nearing the
main sequence, accretion may fall off enough for the stellar atmosphere to become
sufficiently stable for diffusion to begin generating peculiar abundances..
The similar strong chemical peculiarities seen in NGC 2244-334 (Bagnulo et al.,
2004) and HD 72106A seem to support the very early appearance of chemical peculiarities. However the magnetic HAeBe stars HD 190073 and HD 104237, with ages
1.2 ± 0.6 Myr and ∼ 2 Myr respectively, both appear to have solar abundances (Acke
& Waelkens, 2004). This suggests that chemical peculiarities may arise between ∼ 2
and ∼ 10 Myr from the birth line. HD 190073 and HD 104237 both display emission
lines, suggesting they may still be accreting, while NGC 2244-334 and HD 72106A do
not. This raises an alternate possibility: if accretion effectively inhibits the buildup of
chemical peculiarities through diffusion, then chemical peculiarities may simply arise
shortly after a star stops accreting. Thus the observation of chemical peculiarities
may provide information about accretion.
The Ap/Bp type chemical peculiarities in such a young star raise the question of
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
156
whether other types of chemical peculiarities can be seen in the pre-main sequence
or on the ZAMS. In particular, are there non-magnetic species of chemically peculiar
stars on the pre-main sequence or zero age main sequence? The underlying mechanism
of diffusion is the same for both magnetic and non-magnetic chemically peculiar stars.
However, the presence of a magnetic field could provide additional stability to the
stellar atmosphere, allowing Ap/Bp style chemical peculiarities to arise earlier than
non-magnetic patterns of chemical peculiarity.
Given the remarkably strong chemical peculiarities observed in HD 72106A, as well
as in the very young star NGC 2244-334 (Bagnulo et al., 2004), it is interesting to
speculate whether chemical peculiarities are initially very strong in young stars (near
the ZAMS) and then decay as stars age. To properly investigate this speculation one
would required detailed chemical abundances, as well as accurate ages, for a large
number of stars.
The particularly short rotation period of HD 72106A, when compared to other
main sequence Ap stars (Abt & Morrell, 1995), or HD 200775A, is intriguing. One
can speculate that this fast rotation rate may be due to the star still being in the
process of magnetic braking (angular momentum transfer from the star to surrounding
material via the stellar magnetic field). Evidence for angular momentum loss in main
sequence Ap stars has been seen in some cases (Wolff, 1981; Kochukhov & Bagnulo,
2006). However, given the lack of emission, HD 72106A appears to have largely cleared
its immediate neighborhood of circumstellar material. Thus if magnetic braking is
ongoing in HD 72106A, it must be a weak effect, and it must proceed slowly over a
long period of time.
Implications for accretion in the presence of a strong globally ordered photospheric
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
157
magnetic field are not clear (Wade et al., 2007; Catala et al., 2007). While accretion
does not appear to be ongoing in HD 72106A, a clear picture of the magnetic geometry
in such a young star could be valuable in diagnosing the presence and extent of
magnetospheric accretion in similar stars on the pre-main sequence. In contrast, HD
72106B displays emission and infrared excess (Oudmaijer et al., 1992), suggesting it
may still be accreting. However we detect no magnetic field in the secondary, down
to ∼ 200 G. Thus, if accretion is occurring in the star, it seems unlikely that it is
mediated by a strong globally organized magnetic field.
To resolve these questions detailed chemical abundance analysis, as well as an
examination of magnetic fields, must be performed on a wide range of HAeBe stars.
Varying ages, amounts of circumstellar material, and rates of accretion should be
studied. Examining stars of different ages could determine at what point in time
chemical peculiarities arise. Examining different levels of accretion and circumstellar
activity could provide constraints on the stability of the atmosphere needed for diffusion to take place. As well, this may hint at the degree of turbulence caused by low
levels of accretion. Studying a large number of HAeBe stars, both with and without
magnetic fields, can demonstrate which species of chemical peculiarity can arise on
the pre-main sequence. Such a study may also provide indirect information about
accretion and mass loss in HAeBe stars.
In an extension of this study, it would be wise to focus on very young clusters of
stars. In these cases, all the member stars formed at approximately the same time
and in approximately the same environment. Thus one can use cluster averages ages,
and in some cases cluster average distances, to eliminate some of the large uncertainties present in this work. In addition, the cluster provides a context for studies of
CHAPTER 5. SUMMARY, DISCUSSION AND CONCLUSIONS
158
individual stars, for example one could compare apparently peculiar chemical abundances to the cluster’s average abundances. Studying cluster members should allow
for more accurate dating of stars, and possibly improve mass determination, allowing
for more precise conclusions to be drawn.
While there is clearly much work to be done in the study of magnetic pre-main
sequence stars, there are potentially great rewards. Such stars provide the opportunity
to examine diffusion and magnetic field formation in ways previously impossible.
Additionally, these stars may supply critical information about the accretion processes
by which stars form, with wide ranging implications. We are only beginning to see
the range of interesting results that pre-main sequence magnetic stars can provide.
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Appendix A
CFHT Observing Proposal
The following is the observing time request submitted to the Canada-France-Hawaii
Telescope by the author of this thesis. The request was for additional ESPaDOnS
observations of the HD 72106 system, with the primary goal of obtaining a precise
rotation period for the HD 72106A. The request was submitted as a joint Canadian
and French proposal. Dr. Evelyne Alecian kindly agreed be the principal investigator
for the French half of the proposal. The author of this thesis wrote both the Canadian
and French versions of the proposal, and was principal investigator for the Canadian
portion of the proposal. The two versions of the observing propsal are virtualy identical, so for brevity only the Canadian version is included. Our request for observing
time was granted, allowing us to obtain the observations of HD 72106A and B from
March 2007 listed in Table 2.1.
171
172
APPENDIX A. CFHT OBSERVING PROPOSAL
Date: September 21, 2007Category: Stars - Individual, Binaries, Clusters
Proposal: C2036
CFHT
OBSERVING TIME REQUEST
Semester: 2007A Agency: Canada
1. Title of the Program (may be made publicly available for accepted proposals):
Investigating the magnetic chemically peculiar HAeBe star HD 72106A
2. Principal Investigator: Colin Folsom
Postal address: Royal Military College, Station Forces P.O. Box 17000 K7K 7B4 Kingston
Fax:
Phone: 613-530-3540
E-mail: [email protected]
3. Co-Investigators:
Evelyne Alecian
Wade Gregg
Claude Catala
John Landstreet
Institute:
Institute:
Institute:
Institute:
Observatoire de Paris
Royal Military College
Observatoire de Paris
University of Western Ontario
E-mail:
E-mail:
E-mail:
E-mail:
[email protected]
[email protected]
[email protected]
[email protected]
4. Summary of the Program (may be made publicly available for accepted proposals):
Recent groundbreaking observations with ESPaDOnS have shown that about 10% of the pre-main sequence
Herbig Ae/Be (HAeBe) stars display strong globally ordered magnetic fields. This suggests that the progenitors of the main sequence magnetic Ap and Bp stars are the newly discovered magnetic HAeBe stars. The first
HAeBe star in which a magnetic field was discovered with ESPaDOnS, HD 72106A, has very recently been
found to possess strong chemical peculiarities, up to 2 dex above solar. The pattern of abundance anomalies
seen, including overabundances of Si, Cr, and Fe, is characteristic of Ap and Bp stars. HD 72106A is the
only HAeBe star known to display chemical peculiarities. Consequently, an investigation of this unique link
between Ap and HAeBe stars is necessary. An accurate rotation period, magnetic field geometry, and surface
abundance map are required. However, due to the limited number of observations and aliasing effects from
the star’s ∼2 day rotation period, it is impossible to determine these properties from current data. Therefore
we propose to observe the star and collect 14 additional spectra. Numerical experiments show that this will
be sufficient to accurately determine the period of HD 72106A, and allow us to determine the magnetic field
geometry and to map the surface abundance distributions of this remarkable star.
5. Summary of the Observing Run Requested:
Instrument
Detector
Moon (d)
ESPaDOnS
EEV1
14
Time Req.
Service/Queue?
0.5 nights
No
6a. Is this a joint proposal? YES
7a. Is this a Thesis Project? YES
Filters
Queue Mode
—
Grisms
Image Quality
Opt. LST
Min. LST
Max. LST
—
08:00
04:00
12:00
6b. If yes, total number of nights or hours requested from all agencies? 1 nights
7b. If yes, indicate supervisor: Wade
8. Special instrument or telescope requirements:
polarimetric mode
9. Scheduling constraints:
Scheduling for optimal phase coverage: three consecutive 1/3 nights
173
APPENDIX A. CFHT OBSERVING PROPOSAL
Page 2
Proposal: C2036
10. Scientific Justification (science background and objectives of the proposed observations: 1 page maximum):
About 10% of main sequence A and B stars display strong globally ordered magnetic fields - these are the
so-called Ap and Bp stars. The origin and evolution of these magnetic fields is currently the subject of
intensive research. An important avenue of this research is the investigation the progenitors of Ap and Bp
stars. Herbig Ae and Be (HAeBe) stars are the pre-main progenitors of the main sequence A and B type
stars. As such, HAeBe stars have intermediate masses and display emission, infrared excess, and tend to
be associated with dust and nebulosity. It has been long suggested that some HAeBe stars may evolve into
magnetic main sequence Ap and Bp stars. If this were the case, it was hoped that there would be some
distinguishing observable feature in HAeBe stars linking them to Ap stars.
Pioneering ESPaDOnS observations in 2005 and 2006 successfully detected magnetic fields in 4 HAeBe stars
(Wade et al. 2005, Catala et al. 2006, Alecian et al. 2006). The detected fields display similar intensities and
geometries to those of Ap stars (Alecian et al. 2006, Folsom et al. 2006). The longitudinal field strengths
detected are on the order of hundreds of gauss, and the geometries are predominantly dipolar. Additionally,
the frequency of magnetic HAeBe stars is similar to the frequency of magnetic Ap and Bp stars in the set of
all A and B stars, around 10% (Wade et al. 2005). This strongly supports the idea that magnetic HAeBe
stars do in fact represent the progenitors of main sequence magnetic Ap and Bp stars.
The HAeBe star HD 72106A was the first of four HAeBe stars in which magnetic fields were detected with
ESPaDOnS (Fig. 1). It is the only magnetic HAeBe that displays clear chemical peculiarities of the type
seen in Ap stars. Specifically, overabundances in Fe, Ti, and Si of 10 times above solar and Cr of 100
times above solar have been inferred (Folsom et al. 2006). Sample spectrum synthesis is shown in Fig. 2.
Additionally, strongly non-uniform surface abundance distributions are inferred for Cr and Fe. Non-uniform
surface abundances are also characteristic of Ap stars. Therefore, HD 72106A represents a unique opportunity
to study the early development of both chemical peculiarities and magnetic fields in intermediate-mass stars.
The HD 72106 system is a close but optically resolved (0.8”) binary. The secondary, HD 72106B, is a
chemically normal A type HAeBe star. A single ESPaDOnS observation, obtained during conditions of very
good seeing, shows no magnetic field in the secondary, with a 168 gauss limit (at 1σ). Abundance analysis
shows solar concentrations of Cr, Fe, Ti, and Ba (Folsom et al. 2006). Additionally, no asymmetries have
been observed in the line profiles of the star, implying a uniform surface distribution of elements. Thus we
have a normal HAeBe star that is contemporaneous and co-eval with a magnetic chemically peculiar HAeBe
star. This scenario can potentially provide strong additional constraints on the formation of magnetic fields
and chemical peculiarities in A and B stars.
Currently 9 observations have been obtained of HD 72106A. Mean photospheric abundances have been calculated, longitudinal magnetic field measurements obtained, and a rotation period estimated. Despite this,
an unambiguous rotation period cannot be determined from the current data. The Hipparcos photometric
variability of HD 72106A is also too weak to produce a definite period. A periodogram based on longitudinal
field measurements produces many acceptable periods around 2 days (Fig. 3). A more sophisticated technique
looking at the variability in each pixel across the Stokes I and V line profiles also fails to produce a unique
period. Although it is clear that the period is close to 2 days, the lack of a precise period presents a major
obstacle, as this is absolutely necessary to determine the magnetic field geometry and to map the surface
abundance distribution, using a technique like Doppler Imaging.
Numerical experiments show that the period ambiguity results directly from the current time sampling (window function) and the near-integer period in days. Further experiments show that an additional 14 observations, distributed over three consecutive nights, will allow us to unambiguously determine a precise rotation
period for HD 72106A (Fig. 3). Moreover, dense spectroscopic coverage is needed to perform Doppler Imaging.
This is the necessary piece of information for us to begin mapping the surface chemical abundance distribution
and modelling the magnetic field of this unique and remarkable star.
174
APPENDIX A. CFHT OBSERVING PROPOSAL
Page 3
Proposal: C2036
11. References (1 page maximum):
Alecian, E., Catala, C., Donati, J.-F., Wade, G. A., Landstreet, J. D., Böhm, T., Bouret, J.-C., Bagnulo, S.,
Folsom, C. P., Silvester, J. 2006, A&A, in preparation
Catala, C., Alecian, E., Donati, J.-F., Wade, G.A., Landstreet, J.D., Böhm, T., Bouret, J.-C., Bagnulo, S.,
Folsom, C. P., Silvester, J. 2006, A&A, submitted
Donati, J.-F., Semel, M., Carter, B. D., Rees, D. E., Collier Cameron, A. 1997, MNRAS, 291, 658
Folsom, C. P., Wade, G. A., Hanes, D. A., Catala, C., Alecian, E., Bagnulo, S., Böhm, T., Bouret, J.-C.,
Donati, J.-F., Landstreet, J. D. 2006, MNRAS in preparation
Wade, G. A., Donati, J.-F., Landstreet, J. D., Shorlin, S. L. S. 2000, MNRAS, 313, 823
Wade, G. A., Drouin, D., Bagnulo, S., Landstreet, J. D., Mason, E., Silvester, J., Alecian, E., Böhm, T.,
Bouret, J.-C., Catala, C., Donati, J.-F. 2005, A&A, 442, L31
Wade, G. A., Bagnulo, S., Drouin, D., Landstreet, J. D., Monin, D., 2006, MNRAS, submitted
175
APPENDIX A. CFHT OBSERVING PROPOSAL
Page 4
Proposal: C2036
12. Figures (all figures must appear on a single page):
1.04
V/I x20
Intensity
1.02
I
1
0.98
-200
-100
0
km/s
100
200
Figure 1: Illustrative mean LSD Stokes I (bottom) and V (top) profiles of HD 72106A. This
observation corresponds to a longitudinal magnetic field value of 391 ± 65 gauss.
Normalized Intensity
1
0.95
Fe
0.9
Fe
Cr
0.85
Cr
Cr
Si
Fe
Fe
Fe
Fe
Cr
Fe
0.8
4615
4620
4630
4625
Wavelength (A)
4635
4640
Figure 2: Example observed spectrum of HD 72106A, in black, and best fit synthetic spectrum,
in green. The major elements contributing to each line have been labeled. The unusually strong
Cr lines are notable.
12
10
χ2
8
6
4
2
0
1.6
1.8
2
Period (days)
2.2
2.4
Figure 3: Periodograms based on longitudinal magnetic field data. A periodogram based on our
current 9 observations is presented in grey. A periodogram including 14 additional proposed
observations, based on numerical experiments with a period of 1.8 days, is presented in black.
These new observations allow a unique, unambiguous rotation period to be determined, 1.8000 ±
0.0001 days. Such an improvement is the primary goal of this proposal.
176
APPENDIX A. CFHT OBSERVING PROPOSAL
Page 5
Proposal: C2036
13. Technical Justification
(provide technical details of the proposed observations; justify the use of the CFHT, the requested instrument configuration,
and the amount of telescope time requested: 1 page maximum):
The aim of the proposed observations is twofold. The primary objective is to accurately determine the
rotation period of the magnetic HAeBe star HD 72106A. The secondary objective of this program is to
acquire additional high-resolution spectra of HD 72106A for the purposes of Doppler Imaging. To accomplish
this we will acquire a series of observations of the spectrum of HD 72106A in circular polarisation. We will
use the circular polarisation induced in the metallic absorption lines of the star by the longitudinal Zeeman
effect to deduce the longitudinal magnetic field in the star. The longitudinal magnetic field strength varies
with the stellar rotation; thus we can use the variation in the longitudinal field strength over the course of our
observations to determine the rotation period of the star. Addition of the left and right circularly polarised
spectra provides the total intensity (Stokes I) spectrum for free, allowing us to perform Doppler Imaging.
It is important to note that, while ideal, it is not necessary to resolve both components of the HD 72106
binary system for these observations. The spectrum of the secondary appears to be stable, with symmetric
line profiles, except for emission in Hα. Thus if we obtain spectra of the combined light from the system,
we can subtract our existing observation of the secondary off of the combined observation, weighted by the
relative luminosities of the components. Thus we can reconstruct the spectrum of the primary if the seeing is
not sufficiently good to separate the primary from the secondary (Folsom et al. 2006).
We will carry out these observations with ESPaDOnS, the new high-resolution spectropolarimeter at CFHT.
Previous observations of HD 72106A and B in 2005 and 2006 show that ESPaDOnS is fully capable of carrying
out the necessary observations for this program.
ESPaDOnS is essentially the only instrument in the world capable of providing both the polarisation and the
high-resolution spectrum observations necessary for this program. The low-resolution spectropolarimeter at
the VLT, FORS1, may be capable of providing the necessary magnetic field data, however ESPaDOnS provides
some major advantages. ESPaDOnS can resolve the rotationally broadened metallic line profiles in HD
72106A, allowing us to detect a magnetic field even when the longitudinal field value is zero (e.g. dipolar fields
at “crossover” phases). Moreover, there has been some concern raised regarding the suitability of FORS1 for
observations of pre-main sequence objects (Wade et al. 2006), thus we can be more confident in the precision
of our results with ESPaDOnS. Further observations with ESPaDOnS also allow us to maintain a more
homogeneous data set when combined with our previous observations of HD 72106A from the same instrument.
Finally and most importantly, observations with FORS1 would require a second observing campaign to acquire
the high-resolution spectra necessary for Doppler mapping. Narval, the copy of ESPaDOnS being installed at
TBL/Pic du Midi, is not suitable for these observations because the target is too far south to be accessible.
We will use the observing procedure described by Wade et al. (2000), with each exposure divided up into 4
subexposures and the quarter wave plate rotated by ±90 degrees between each subexposure. The data reduction will be performed with Libre-ESpRIT (Donati et al. 1997; Donati et al., in preparation). If observations
of the individual components of the system are not possible due to the seeing conditions, the spectrum of the
primary will be reconstructed. Least-Squares Deconvolution (Donati et al. 1997) and complementary techniques will be used to determine longitudinal magnetic field values. The rotation period will then be found
by fitting models thorough the magnetic field data and constructing a periodogram, and by direct modelling
of Stokes V profiles. Doppler Imaging will be performed using the code “Inverse” with which we and our
collaborators have past expertise. All the members of our team are experts at data reduction and analysis of
spectropolarimetric observations.
Based on our previous observations of HD 72106A, to achieve a peak S/N ratio of about 250 we will require
an exposure time of 600x4 seconds. Including a 160 second readout time and an additional 30 seconds of
overhead we will require 2590 seconds per exposure. Thus we can acquire the necessary 14 observations in 10
hours, or about one night, However, for optimal phase coverage, we request that this be distributed uniformly
over 3 consecutive nights, and spread evenly between the Canadian and French agencies.
177
APPENDIX A. CFHT OBSERVING PROPOSAL
Page 6
Proposal: C2036
14. Targets:
Object/Field
HD 72106
α
08:29:34.9
δ
-38:36:21.1
Epoch
2000
Mag/Flux
8.58
Comment
15. General Target Information:
The target details represent both components of this 0.8” separation binary system. The V magnitude of the
primary, our principal object of interest, is 9.00 and the magnitude of the secondary is 9.62. To obtain a peak
S/N ratio of 250 in the primary we require a 600x4 second exposure.
16. Is this program conducted in relation with other observations (optical, radio, space)?
YES: This program follows up earlier discovery observations obtained with ESPaDOnS.
17. How many additional nights or hours at CFHT would be required to complete this project? 0 nights
18a. Is an extension of the one-year proprietary period required? NO
18c. If yes, justify the request for an extension:
18b. Proposed proprietary period? 12 months
178
APPENDIX A. CFHT OBSERVING PROPOSAL
Page 7
Proposal: C2036
19. Recent Allocations on CFHT and Other Telescopes:
1. CFHT, 2005A, 2005B, 6 nights, (Wade et al.), “Seeking the progenitors of magnetic Ap stars”, CFHT,
2005A, 2005B, 7 nights, (Catala et al.), “Magnetic fields in the pre-main sequence Herbig Ae/Be stars”,
observations obtained for approximately 60 Herbig Ae/Be stars, data fully reduced, magnetic fields detected
in 4 stars (modelling is in progress). One paper has been published (Wade et al., 2005, A&A, 442, L31) + 3
papers (Catala et al. 2006, Alecian et al. 2006, Folsom et al. 2006) and 3 theses (2 MSc - C. Folsom and K.
Bale, 1 PhD - E. Alecian) in preparation.
2. CFHT, 2005A, 3 nights, (Landstreet et al.), “A spectropolarimetric survey of magnetic stars in open
clusters”, Lots of stars observed, a number of new magnetic field detections and confirmations. Data reduction
was performed at CFHT and subsequent magnetic field measurements were made. We are incorporating these
data into a cluster database and they will appear in a forthcoming paper. (Thesis of an undergraduate student)
3. CFHT, 2005B, 2 nights, (Wade et al), “Magnetic fields of massive stars in the ONC”, data fully reduced
and currently being analyzed, magnetic field detected in at least one star. (PhD thesis - V. Petit)
4. CFHT, 2005B, 1.5 nights, (Wade et al.), “Magnetic Doppler Imaging of Ap stars”, no program data
acquired due to instrument problems, excellent backup program data acquired which is now fully reduced and
under analysis.
5. CFHT, 2006B, 3 nights, (Catala et al.), “Magnetism of pre-main sequence A and B stars in young open
clusters: the influence of environment age and rotation”, the first part of a multi-semester project. 1 night
already secured in Aug. 2006, 8 stars observed, 1 magnetic field detection; 2 nights remaining in Dec. 2006.
20. Publications Resulting from CFHT Observations (only the 12 most recent contained in the database are displayed):
Donati, J.-F., Howarth, I. D., Bouret, J.-C., Petit, P., Catala, C., Landstreet, J. 2006, MNRAS, 365, L6
C. Catala, E. Alecian, J.-F. Donati, G.A. Wade, J.D. Landstreet, T. Böhm, J.-C. Bouret, S. Bagnulo, C.
Folsom, J. Silvester 2006, A&A, submitted
Wade, G.A., Drouin, D., Bagnulo, S., Landstreet, J.D., Mason, E., Silvester, J., Alecian, E., Böhm, T.,
Bouret, J.-C., Catala, C., Donati, J.-F. 2005, A&A, 442, L31
Unruh, Y. C. and 23 others 2004, MNRAS, 348, 1301
Kochukhov, O., Ryabchikova, T., Landstreet, J. D. , Weiss, W. W. 2004, MNRAS, 351, L34
Kochukhov, O., Landstreet, J. D., Ryabchikova, T., Weiss, W. W., Kupka, F. 2002, MNRAS, 337, L1
Sigut, T. A. A., Landstreet, J. D. and Shorlin, S. L. S. 2000, ApJ, 530, L89
Catala, C. and 26 others 1999, A&A, 345, 884
Landstreet, J.D. 1998, A&A, 338, 1041
Hill, G. M., Bohlender, D. A., Landstreet, J. D., Wade, G. A., Manset, N., Bastien, P. 1998, MNRAS, 297,
236
Ryabchikova, T. A., Landstreet, J. D., Gelbmann, M. J., Bolgova, G. T., Tsymbal, V. V. and Weiss, W. W.
1997, A&A, 327, 1137
Mathys, G., Hubrig, S., Landstreet, J. D., Lanz, T. and Manfroid, J. 1997, A&AS, 123, 353
Disclaimer: In submitting this application, I acknowledge that I am aware of CFHT’s policy concerning public access to
data after a proprietary period of one year. I recognize that each individual reacts differently to working at high altitude and
that some individuals may experience potentially severe altitude sickness or other medical problems. I agree that observers
proposing to work at Mauna Kea should be medically fit for such work and not have conditions which would be inconsistent
with work at high altitude. I understand and agree that Canada-France-Hawaii Telescope Corporation and those acting in its
behalf have no liability with respect to the risks associated with work at the telescope by observers or others, and that every
participant in an observing run at Mauna Kea should follow the policy of his or her own employer or sponsoring agency with
respect to medical examinations and other requirements for work at high altitude.
Signature: signed via “POOPSY”
Form generated by CADC/HIA/NRC with “POOPSY”
Appendix B
Computer Programs Written
Included here are the raw Fortran source codes use for the continuum normalization,
LSD profile period searching, and spectrum fitting routines. Hard-coded parameters
are set to typical values used in this thesis.
B.1
Continuum Normalization: norm.f
The continuum normalization routine used for all spectra in this thesis. The routine
fits a low order polynomial through carefully selected points in the continuum of
the star, then divides the spectrum by the polynomial. This is done for each order
individually. Points in the continuum are selected by finding the pixel with the most
flux in a large segment of spectrum, once a running average has been applied to the
spectrum.
c
c
c
c
c
c
c
c
Originally from Kathryn Bale (April 2006)
Heavily Modified by Colin Folsom (May 2006)
Colin’s Change Log:
Added flexibility for slightly different sized files
(uses over sized arrays and the number of lines in the file header)
Fixed the read in, uncertainty (rmaxdy), and write out to include the
uncertainty column from ESPADONS spectra
179
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c
c
c
c
c
c
c
c
c
c
c
c
c
c
Capped the output values at +/- 10 for easy reading.
Now takes a running average of several (17) points and finds the highest point in a large
(200 point) bin. These points are used to fit the polynomial (this should give points
that are ~ noise free and not in absorption lines)
Uncertainties for the points are calculated accordingly.
Changed the excluded regions to read use the highest point in this bin, not the last
(important for larger bins).
Now includes the ’balmer’ subroutine which looks ahead/back to the far side of a
Balmer line to fit the continuum in this region
(assumes that you will use a low order (2nd) polynomial)
If the calibration between orders is good it gives a straight line across the Balmer line,
and hence a ’1st order’ approximation of the continuum
Uses 5 points on the far side found with a copy of the regular point finding code
(if you change the regular point finding routine change this too!)
C
C
My apologies to the reader for the legacy goto statements!
I’ll clean it up soon, I promise!
implicit real*8(a-h,o-z)
implicit integer(i-n)
character*15 filenm
character*50 junk1,junk2
dimension rstart(1),rend(1),fn(9)
dimension wl(220000),r1(220000),r2(220000),r3(220000),r4(220000),
1
rn(220000),x(2000),y(2000),dy(2000),coeff1(10),coeff2(10),
2
r5(220000)
dimension nos(40),noe(40),ff(40),coeff3(10)
character*2 fn
character*2 ff
data ff/’01’,’02’,’03’,’04’,’05’,’06’,’07’,’08’,’09’,
1
’10’,’11’,’12’,’13’,’14’,’15’,’16’,’17’,’18’,’19’,
2
’20’,’21’,’22’,’23’,’24’,’25’,’26’,’27’,’28’,’29’,
3
’30’,’31’,’32’,’33’,’34’,’35’,’36’,’37’,’38’,’39’,’40’/
c Continuum normalise Esprit-reduced spectra using polynomial fit.
c This version selectes continuum points from x40x pixel bins for the fit,
c useful for rich-spectrum stars.
mode=+1
nclips=1
100
c
format(a15)
read input filename (now hard coded... was a loop here reading from the file flistE)
filenm = ’in.s’
open(10,file=’N’//filenm,status=’unknown’)
7 is the file containing the unnormalized spectrum
open(7,file=filenm,status=’old’)
c
read(7,101) junk1
read(7,102) iflen, ncol
write(10,101) junk1
write(10,102) iflen, ncol
101
102
format(a50)
format(I7,1X,I1)
c
numorder = the order that we are on, initially one
numorder=1
c
nos = the start line for each order (first one starts at one)
nos(numorder)=1
c
finds the beginning and ending lines of each order
do i=1,iflen
read(7,1105) wl(i),r1(i),r2(i),r3(i),r4(i), r5(i)
if the space between the wavelengths is bigger than 0.2nm
if (i.ne.1.and.(wl(i)-wl(i-1))**2.gt.0.04) then
write(*,*) ’Order: ’, numorder, ’, Wavelength: ’, wl(i)
noe is the line the order ends on
c
c
180
APPENDIX B. COMPUTER PROGRAMS WRITTEN
noe(numorder)=i-1
numorder=numorder+1
nos(numorder)=i
end if
end do
noe(numorder)=iflen
write(*,*) ’There are ’,numorder,’ orders.’
close(7)
do i=1,numorder
c
write to the files numbered 01 to 40
open(14,file=ff(i),status=’unknown’)
c
npts is the total number of points in the order
npts=0
rmaxy=-1000.0
c
set the number of points per bin
nbin=200
c--------------------------------write(*,*) i, nbin
c--------------------------------c
initialize counters for the number of pointes on the edges of the Balmer lines
c
necessary for the blamer subroutine
iHbeta1 = 0
iHbeta2 = 0
iHalpha1 = 0
iHalpha2 = 0
iHgamma1 = 0
iHgamma2 = 0
iHdelta1 = 0
iHdelta2 = 0
c
c
Throw out the first x20 and last x20 points in the order to get a
better fit. Then pick the highest point in each set of x40 points.
do k=nos(i)+40,noe(i)-40
c
17 pt running average
ravg = 0.0
ravgdy = 0.0
do j=-8,8
ravg =ravg +r1(k+j)
ravgdy = ravgdy + r5(k+j)*r5(k+j)
enddo
ravg=ravg/17.0
ravgdy = sqrt(ravgdy)/17.0
c
if it’s the highest point so far
if(ravg.gt.rmaxy) then
rmaxy=ravg
rmaxx=wl(k)-wl(nos(i))
rmaxdy=ravgdy
end if
c
c
if it’s evenly divisible by nbin, hence every nbinth point
if(real(k/nbin).eq.real(k)/real(nbin)) then
get rid of Balmer lines and other large absorption lines
if(rmaxx+wl(nos(i)).gt.371.8.and.
1
rmaxx+wl(nos(i)).lt.372.5) goto 91
if(rmaxx+wl(nos(i)).gt.372.9.and.
1
rmaxx+wl(nos(i)).lt.374.0) goto 91
if(rmaxx+wl(nos(i)).gt.374.5.and.
1
rmaxx+wl(nos(i)).lt.375.9) goto 91
181
APPENDIX B. COMPUTER PROGRAMS WRITTEN
if(rmaxx+wl(nos(i)).gt.376.3.and.
rmaxx+wl(nos(i)).lt.378.1) goto
if(rmaxx+wl(nos(i)).gt.378.6.and.
rmaxx+wl(nos(i)).lt.380.6) goto
if(rmaxx+wl(nos(i)).gt.381.7.and.
rmaxx+wl(nos(i)).lt.382.2) goto
if(rmaxx+wl(nos(i)).gt.382.3.and.
rmaxx+wl(nos(i)).lt.385.1) goto
if(rmaxx+wl(nos(i)).gt.387.0.and.
rmaxx+wl(nos(i)).lt.391.0) goto
if(rmaxx+wl(nos(i)).gt.395.0.and.
rmaxx+wl(nos(i)).lt.400.0) goto
if(rmaxx+wl(nos(i)).gt.400.5.and.
rmaxx+wl(nos(i)).lt.401.3) goto
if(rmaxx+wl(nos(i)).gt.402.1.and.
rmaxx+wl(nos(i)).lt.403.1) goto
1
1
1
1
1
1
1
1
c
91
91
91
91
91
91
91
91
H delta
if(rmaxx+wl(nos(i)).gt.405.2.and.
rmaxx+wl(nos(i)).lt.414.0)then
wlstart = 405.2
wlend = 414.0
call balmer(wlstart, wlend, iHdelta1, iHdelta2, i,
k, nos, noe, wl, r1, r5, rmaxy, rmaxx, rmaxdy)
if(rmaxy.eq.-1000.0) goto 91
endif
1
2
if(rmaxx+wl(nos(i)).gt.417.0.and.
rmaxx+wl(nos(i)).lt.417.5) goto 91
if(rmaxx+wl(nos(i)).gt.419.8.and.
rmaxx+wl(nos(i)).lt.421.2) goto 91
1
1
c
H gamma
if(rmaxx+wl(nos(i)).gt.428.0.and.
rmaxx+wl(nos(i)).lt.438.0)then
wlstart = 428.0
wlend = 438.0
call balmer(wlstart, wlend, iHgamma1, iHgamma2, i,
k, nos, noe, wl, r1, r5, rmaxy, rmaxx, rmaxdy)
if(rmaxy.eq.-1000.0) goto 91
endif
1
2
c
H beta
if(rmaxx+wl(nos(i)).gt.478.5.and.
rmaxx+wl(nos(i)).lt.494.5)then
wlstart = 478.5
wlend = 494.5
call balmer(wlstart, wlend, iHbeta1, iHbeta2, i,
k, nos, noe, wl, r1, r5, rmaxy, rmaxx, rmaxdy)
if(rmaxy.eq.-1000.0) goto 91
endif
1
2
if(rmaxx+wl(nos(i)).gt.501.3.and.
rmaxx+wl(nos(i)).lt.502.6) goto
if(rmaxx+wl(nos(i)).gt.516.6.and.
rmaxx+wl(nos(i)).lt.517.4) goto
if(rmaxx+wl(nos(i)).gt.587.0.and.
rmaxx+wl(nos(i)).lt.590.0) goto
if(rmaxx+wl(nos(i)).gt.626.9.and.
rmaxx+wl(nos(i)).lt.629.4) goto
1
1
1
1
c
91
91
91
91
H alpha
1
2
1
1
1
if(rmaxx+wl(nos(i)).gt.643.5.and.
rmaxx+wl(nos(i)).lt.669.5)then
wlstart = 643.5
wlend = 669.5
call balmer(wlstart, wlend, iHalpha1, iHalpha2, i,
k, nos, noe, wl, r1, r5, rmaxy, rmaxx, rmaxdy)
if(rmaxy.eq.-1000.0) goto 91
endif
if(rmaxx+wl(nos(i)).gt.686.5.and.
rmaxx+wl(nos(i)).lt.695.2) goto 91
if(rmaxx+wl(nos(i)).gt.759.2.and.
rmaxx+wl(nos(i)).lt.766.8) goto 91
if(rmaxx+wl(nos(i)).gt.769.2.and.
rmaxx+wl(nos(i)).lt.771.5) goto 91
182
APPENDIX B. COMPUTER PROGRAMS WRITTEN
if(rmaxx+wl(nos(i)).gt.775.8.and.
rmaxx+wl(nos(i)).lt.778.5) goto
if(rmaxx+wl(nos(i)).gt.812.7.and.
rmaxx+wl(nos(i)).lt.814.0) goto
if(rmaxx+wl(nos(i)).gt.813.8.and.
rmaxx+wl(nos(i)).lt.814.6) goto
if(rmaxx+wl(nos(i)).gt.816.0.and.
rmaxx+wl(nos(i)).lt.816.7) goto
if(rmaxx+wl(nos(i)).gt.818.2.and.
rmaxx+wl(nos(i)).lt.819.1) goto
if(rmaxx+wl(nos(i)).gt.822.6.and.
rmaxx+wl(nos(i)).lt.851.8) goto
if(rmaxx+wl(nos(i)).gt.839.1.and.
rmaxx+wl(nos(i)).lt.839.9) goto
if(rmaxx+wl(nos(i)).gt.843.3.and.
rmaxx+wl(nos(i)).lt.845.0) goto
if(rmaxx+wl(nos(i)).gt.844.9.and.
rmaxx+wl(nos(i)).lt.845.4) goto
if(rmaxx+wl(nos(i)).gt.847.9.and.
rmaxx+wl(nos(i)).lt.850.8) goto
if(rmaxx+wl(nos(i)).gt.853.3.and.
rmaxx+wl(nos(i)).lt.880.1) goto
if(rmaxx+wl(nos(i)).gt.857.7.and.
rmaxx+wl(nos(i)).lt.862.5) goto
if(rmaxx+wl(nos(i)).gt.863.5.and.
rmaxx+wl(nos(i)).lt.870.7) goto
if(rmaxx+wl(nos(i)).gt.871.8.and.
rmaxx+wl(nos(i)).lt.878.8) goto
if(rmaxx+wl(nos(i)).gt.881.7.and.
rmaxx+wl(nos(i)).lt.890.4) goto
if(rmaxx+wl(nos(i)).gt.894.3.and.
rmaxx+wl(nos(i)).lt.911.0) goto
if(rmaxx+wl(nos(i)).gt.899.8.and.
rmaxx+wl(nos(i)).lt.903.3) goto
if(rmaxx+wl(nos(i)).gt.916.6.and.
rmaxx+wl(nos(i)).lt.926.9) goto
if(rmaxx+wl(nos(i)).gt.927.2.and.
rmaxx+wl(nos(i)).lt.941.3) goto
if(rmaxx+wl(nos(i)).gt.946.5.and.
rmaxx+wl(nos(i)).lt.957.8) goto
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
npts=npts+1
x(npts)=rmaxx
y(npts)=rmaxy
dy(npts)=rmaxdy
91
rmaxy=-1000.0
end if
end do
c
c
c
Fit...
do ik=1,10
coeff1(ik)=0.
end do
c
set the number of terms used in polyfit
nterms=4
if(i.eq.1)
if(i.eq.2)
if(i.eq.3)
if(i.eq.4)
if(i.eq.5)
if(i.eq.6)
if(i.eq.7)
if(i.eq.8)
if(i.eq.9)
nterms=2
nterms=3
nterms=2
nterms=2
nterms=2
nterms=2
nterms=2
nterms=5
nterms=2
91
91
91
91
91
91
91
91
91
91
91
91
91
91
91
91
91
91
91
91
183
APPENDIX B. COMPUTER PROGRAMS WRITTEN
if(i.eq.10)
if(i.eq.11)
if(i.eq.12)
if(i.eq.13)
if(i.eq.14)
if(i.eq.15)
if(i.eq.16)
if(i.eq.17)
if(i.eq.18)
if(i.eq.19)
if(i.eq.20)
if(i.eq.21)
if(i.eq.22)
if(i.eq.23)
if(i.eq.24)
if(i.eq.25)
if(i.eq.26)
if(i.eq.27)
if(i.eq.28)
if(i.eq.29)
if(i.eq.30)
if(i.eq.31)
if(i.eq.32)
if(i.eq.33)
if(i.eq.34)
if(i.eq.35)
if(i.eq.36)
if(i.eq.37)
if(i.eq.38)
if(i.eq.39)
if(i.eq.40)
nterms=2
nterms=5
nterms=5
nterms=5
nterms=5
nterms=2
nterms=2
nterms=5
nterms=6
nterms=5
nterms=5
nterms=5
nterms=5
nterms=5
nterms=5
nterms=5
nterms=5
nterms=2
nterms=2
nterms=5
nterms=5
nterms=5
nterms=5
nterms=3
nterms=2
nterms=2
nterms=2
nterms=2
nterms=2
nterms=3
nterms=3
call polyfit(x,y,dy,npts,nterms,mode,coeff1,chisq)
c
c
c
output the renormalized values
do k=nos(i),noe(i)
calculate the value of the fit polynomial at this point
rfit1=0.
nhow=nterms
do nfit=1,nhow
rfit1=rfit1+coeff1(nfit)*(wl(k)-wl(nos(i)))**(nfit-1)
end do
then apply the polynomial and write the result !!!Limiting the extrema to +/-10!!!
if( (r1(k)/rfit1).gt.10.0 )then
write(10,1105) wl(k),10.0,10.0
1
,10.0,10.0, 10.0
write(14,1051) wl(k),r1(k),rfit1
elseif( (r1(k)/rfit1).lt.-10.0 )then
write(10,1105) wl(k),-10.0,-10.0
1
,-10.0,-10.0, -10.0
write(14,1051) wl(k),r1(k),rfit1
else
write(10,1105) wl(k),r1(k)/rfit1,r2(k)/r1(k)
1
,r3(k)/r1(k),r4(k)/r1(k), r5(k)/rfit1
write(14,1051) wl(k),r1(k),rfit1
endif
end do
close(14)
c
(end the loop over orders)
end do
close(10)
close(20)
close(21)
close(22)
1051 format(f11.4,1x,3(f12.5,1x))
1105 format(f10.4,1x,5(e11.4,1x))
184
APPENDIX B. COMPUTER PROGRAMS WRITTEN
end
subroutine polyfit(x,y,sigmay,npts,nterms,mode,a,chisqr)
implicit real*8(a-h,o-z)
implicit integer(i-n)
dimension x(1),y(1),sigmay(1),a(1)
dimension sumx(19),sumy(10),array(10,10)
c
c From Bevington 1st edition
c
c
Accumulate weighted sums
c
11
13
15
21
31
32
33
35
37
39
41
44
45
48
49
50
nmax=2*nterms-1
do 13 n=1,nmax
sumx(n)=0.
do 15 j=1,nterms
sumy(j)=0.
chisq=0.
do 50 i=1,npts
xi=x(i)
yi=y(i)
if (mode) 32, 37, 39
if (yi) 35,37,33
weight=1./(yi)
go to 41
weight=1./(-yi)
go to 41
weight=1.
go to 41
weight=1./sigmay(i)**2
xterm=weight
do 44 n=1,nmax
sumx(n)=sumx(n)+xterm
xterm=xterm*xi
yterm=weight*yi
do 48 n=1,nterms
sumy(n)=sumy(n)+yterm
yterm=yterm*xi
chisq=chisq+weight*yi**2
continue
c
c Construct matrices and calculate components
c
51
do 54 j=1,nterms
do 54 k=1,nterms
n=j+k-1
54
array(j,k)=sumx(n)
delta=determ(array,nterms)
if(delta) 61,57,61
57
chisqr=0.
do 59 j=1,nterms
59
a(j)=0.
go to 80
61
do 70 l=1,nterms
62
do 66 j=1,nterms
do 65 k=1,nterms
n=j+k-1
65
array(j,k)=sumx(n)
66
array(j,l)=sumy(j)
70
a(l)=determ(array,nterms)/delta
c
185
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c Calculate chi**2
c
71
do 75 j=1,nterms
chisq=chisq-2.*a(j)*sumy(j)
do 75 k=1,nterms
n=j+k-1
75
chisq=chisq+a(j)*a(k)*sumx(n)
76
free=npts-nterms
77
chisqr=chisq/free
80
return
end
function determ(array,norder)
implicit real*8(a-h,o-z)
implicit integer(i-n)
dimension array(10,10)
c
c Calculate the determinant of a square matrix
c
110
111
determ=1.
do 150 k=1,norder
if(array(k,k)) 141,121,141
do 123 j=k,norder
if(array(k,j)) 131,123,131
continue
determ=0.
go to 160
do 134 i=k,norder
save=array(i,j)
array(i,j)=array(i,k)
array(i,k)=save
determ=-determ
determ=determ*array(k,k)
if (k-norder) 143,150,150
k1=k+1
do 146 i=k1,norder
do 146 j=k1,norder
array(i,j)=array(i,j)-array(i,k)*array(k,j)/array(k,k)
continue
return
end
121
123
131
134
141
143
146
150
160
C
C
C
subroutine balmer(wlstart, wlend, iH1, iH2, i, k,
nos, noe, wl, r1, r5, rmaxy, rmaxx, rmaxdy)
Looks ahead/behind to the far side of the Balmer line to finde a few (5) good points.
returns a good point per call, untill iH1 = 5 (for the blue side)
and iH2 = 5 (for the red side)
c
c
c
c
c
c
c
c
wlstart, wlend = starting and ending wavelengths of the balmer line
i, k = the order and point we’re currently on
iH1, iH2 = the number of points at the start and end of the balmer line use so far
(gets modified) (goes up to 5 total)
wl, r1, r5 = wavelength, intensity, and uncertainty arrays
rmaxy, rmaxx, rmaxdy = vales fot the best point in this bin (200 point bins)
(gets modified and returned)
returns rmaxy = -1000.0 if we are on more then the 5th point
2
implicit real*8(a-h,o-z)
implicit integer(i-n)
2
c
c
real*8 wlstart, wlend, wl(220000), r1(220000),
r5(220000), rmaxy, rmaxx, rmaxdy
integer i, k, iH1, iH2, nos(40), noe(40)
if(iH1.lt.5.and.iH2.lt.5)then
if this order is on the blue side
if(wl(nos(i)).lt.wlstart)then
advance past the balmer line
jend=k
186
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c
c
c
1
c
c
c
c
c
187
do while (wl(jend).lt.wlend)
jend=jend+1
enddo
and skipp ahead iH1 bins
jend=jend+200*iH1
iH1 = iH1+1
if this order is on the red side
elseif (wl(noe(i)).gt.wlend)then
rewinde past the balmer line
jend=k
do while (wl(jend).gt.wlstart)
jend=jend-1
enddo
iH2 = iH2+1
and step back iH2 bins (starting at -200)
jend=jend-200*iH2
endif
search for best point (out of 200) with parameters stolen from the body of this code
rmaxy=-1000.0
do j=jend,jend+200
17 pt running average
ravg = 0.0
do j2=-8,8
ravg=ravg +r1(j+j2)
enddo
ravg=ravg/17.0
if it’s the highest point so far keep it
if(ravg.gt.rmaxy) then
rmaxy=ravg
rmaxx=wl(j)-wl(nos(i))
rmaxdy=r5(j)
end if
enddo
if we have enough points just skipp all this and return -1000
else
rmaxy=-1000.0
endif
end
B.2
Period Searching with Stokes I and V LSD
Profiles: pbp.f
This routine searches for periodicity in the Stokes I and V LSD profiles independently.
Sinusoids with a range of periods are fit through the time series of observations at
one particular pixel in the LSD profile, and the reduced χ2 is calculated. This is
then repeated for each pixel individually. The reduced χ2 values for each period are
averaged over all pixels, and a periodogram (plot of reduced χ2 versus period) is
output.
APPENDIX B. COMPUTER PROGRAMS WRITTEN
implicit none
LSD version - cleaned up a bit (April 4 2007)
Things that may change: nLines, nFiles, vmin, vmax, read format,
individual pixel writing (currently #100)
See ./Test/pbpt.f for the version with multiple pixel phases output
Interpolation Added. Uses the 1st input LSD profile wavelenght scale
c
c
C
c
C
integer iFLenght, nColumn, n, nLines, nFiles, nDates, i, ipNum,
iminI, iminV, iFito, iFitod, iCount, pix1, pix2
character*45 txt
double precision period0, jd0, pMin, pMax, vmin, vmax,
c
fminI, fminV, limI, limV, sign, maxVarIT, maxVarVT, wtI, wtV
c
c
set the number of lines in the input spectra (excluding header)
parameter(nLines=188)
set the number of observations (points in the time series)
parameter(nFiles=18)
set the start and end of the line in velocity units relative to the line
center in the star’s frame
parameter(vmin = -30.0)
parameter(vmax = 70.0)
set the order of the fit polynomial
parameter(iFito = 1)
parameter(pix1 = 57)
parameter(pix2 = 66)
pix2 = pix1 +10 (start and end pixels for the list of phased I and V curves)
c
c
c
c
c
character*40 fnames(nFiles)
double precision wl(nFiles,nLines),int(nFiles,nLines),
c
eint(nFiles,nLines), vi(nFiles,nLines), evi(nFiles,nLines),
c
no(nFiles,nLines), eno(nFiles,nLines), JD(nFiles),
c
dint(nFiles), deint(nFiles), dvi(nFiles), devi(nFiles),
c
maxVarI(nLines), maxVarV(nLines), avgI(nLines), avgV(nLines),
c
tint(nLines), teint(nLines), tvi(nLines), tevi(nLines)
c
real, allocatable :: pspecI(:,:), pspecIT(:,:),
pspecVT(:,:), phasesI(:,:), phasesV(:,:)
pspecV(:,:),
avgI = 0.0
avgV = 0.0
wtV = 0.0
wtI = 0.0
C
Read in the input files
c
read the 1st line of flist...
open(20, file="flist", status=’unknown’)
99
c
c
c
c
54
c
c
read(20,99) nDates, period0, jd0
format(I4, 1X, F10.4, 1X, F16.4)
if(nDates.ne.nFiles)then
write(*,*) ’Warning: number of files inconsistant with the nFile
*s parameter’
endif
loop over all observations
do n=1,nFiles
read the flist line corrisponding to this observation
read(20,103) JD(n), fnames(n)
read the observed LSD file
open(21, file=fnames(n), status=’unknown’)
comment out these two line if there is no headder on the LSD profiles
read(21,*)
read(21,54) iFLenght, nColumn
format(I4,I2)
do i=1,nLines
read(21,102) wl(n,i), int(n,i), eint(n,i), vi(n,i),
c
evi(n,i), no(n,i), eno(n,i)
while we’re looping through pixels, lets find the average value in I and V for this pixel
(just a sum weighted by 1/uncertainty for now...)
188
APPENDIX B. COMPUTER PROGRAMS WRITTEN
avgI(i) = avgI(i)+int(n,i)/eint(n,i)
avgV(i) = avgV(i)+vi(n,i)/evi(n,i)
enddo
close(21)
enddo
read the last line of flist
read(20,104) pMin, pMax, ipNum
format(F10.4,F10.4,I10)
c
104
close(20)
102
c
103
format(F10.4,1X,E11.4,1X,E11.4,1X,E11.4,1X,E11.4,1X,E11.4,1X,
E11.4)
format(F13.5,1X,A40)
c
allocate the result arrays
ALLOCATE(pspecI(ipNum,2),pspecIT(ipNum,2))
ALLOCATE(pspecV(ipNum,2),pspecVT(ipNum,2))
ALLOCATE(phasesI(nFiles,ipNum),phasesV(nFiles,ipNum))
c
initialize the arrays to 0 (aren’t array operations fun?)
pspecIT = 0.0
pspecI = 0.0
pspecVT = 0.0
pspecV = 0.0
iCount = 0
maxVarIT = 0.0
maxVarVT = 0.0
maxVarI = 0.0
maxVarV = 0.0
tint=0.0
c
Interpolate to common wavelenght points, use the 1st LSD profile
do i=2, nFiles
call interp(wl(1,:), int(1,:), nLines, wl(i,:), int(i,:),
1
nLines, tint)
int(i,:) = tint
1
call interp(wl(1,:), eint(1,:), nLines, wl(i,:), eint(i,:),
nLines, tint)
eint(i,:) = tint
1
call interp(wl(1,:), vi(1,:), nLines, wl(i,:), vi(i,:),
nLines, tint)
vi(i,:) = tint
1
call interp(wl(1,:), evi(1,:), nLines, wl(i,:), evi(i,:),
nLines, tint)
evi(i,:) = tint
wl(i,:) = wl(1,:)
enddo
c
c
c
then find the maximum variation in I and V
do i=1,nLines
but first finish calculating that average
(devide by sum of weighting (ie sum of 1/uncertainty))
wtI = 0.0
wtV = 0.0
do n=1,nFiles
wtI = wtI + 1/eint(n,i)
wtV = wtV + 1/evi(n,i)
189
APPENDIX B. COMPUTER PROGRAMS WRITTEN
enddo
avgI(i) = avgI(i)/wtI
avgV(i) = avgV(i)/wtV
c
Now look for the observation with the largest _sigma_ differance from the mean
do n=1,nFiles
sign = ABS(int(n,i)-avgI(i))/eint(n,i)
if(sign.gt.maxVarI(i))then
maxVarI(i)=sign
endif
sign = ABS(vi(n,i)-avgV(i))/evi(n,i)
if(sign.gt.maxVarV(i))then
maxVarV(i)=sign
endif
c
enddo
And the largest sigma differance of all
if(maxVarI(i).gt.maxVarIT)then
maxVarIT = maxVarI(i)
endif
if(maxVarV(i).gt.maxVarVT)then
maxVarVT = maxVarV(i)
endif
enddo
wtV = 0.0
wtI = 0.0
c
loop over all pixels in the LSD spectrum
do i=1,nLines
write(*,*) ’On pixel: ’, i
if( (wl(1,i).gt.vmin).and.(wl(1,i).lt.vmax) )then
iCount = iCount+1
write(*,*) ’Pixels included: ’, iCount
c
use a copy of the intensity and error (error gets modified in the subroutine)
dint = int(:,i)
deint = eint(:,i)
iFitod = iFito
********************************************************************
call fsrch(nFiles, period0, jd0, iFitod, JD, dint,
1
deint, pMin, pMax, ipNum, pspecI, phasesI,
2
iCount)
********************************************************************
c
c
Weighted average, by the variation(/uncertainty) of the point
as a fraction of the largest variation of any point
pspecIT = pspecIT + pspecI*(maxVarI(i)/maxVarIT)
wtI = wtI+(maxVarI(i)/maxVarIT)
c
And do the same for stokes V
dvi = vi(:,i)
devi = evi(:,i)
iFitod = iFito
********************************************************************
call fsrch(nFiles, period0, jd0, iFitod, JD, dvi,
1
devi, pMin, pMax, ipNum, pspecV, phasesV,
2
iCount)
********************************************************************
c
c
Weighted average, by the varyation(/uncertainty) of the point
as a fraction of the largest variation of any point
pspecVT = pspecVT + pspecV*(maxVarV(i)/maxVarVT)
wtV = wtV+(maxVarV(i)/maxVarVT)
endif
enddo
190
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c
Divide by the sum of the weighting
pspecIT = pspecIT/wtI
pspecVT = pspecVT/wtV
c
Finaly parse and output the results
fminI = pspecIT(1,2)
iminI = 0
fminV = pspecVT(1,2)
iminV = 0
open(31, file="avg_periodsI", status=’unknown’)
open(32, file="avg_periodsV", status=’unknown’)
loop over all lines
do i=1,ipNum
write the ones that are not zero (skip the remainder)
if(pspecIT(i,2).ne.0.0)then
write(31,200) pspecIT(i,1), pspecIT(i,2)
and find the minimum chi^2 point
if(pspecIT(i,2).lt.fminI)then
fminI = pspecIT(i,2)
iminI = i
endif
endif
and do the same for the stokes V spectrum too
if(pspecVT(i,2).ne.0.0)then
write(32,200) pspecVT(i,1), pspecVT(i,2)
if(pspecVT(i,2).lt.fminV)then
fminV = pspecVT(i,2)
iminV = i
endif
endif
enddo
format(F13.7,1X,E16.8)
close(31)
close(32)
c
c
c
c
200
C
Final comments written to terminal
c
write the phases of the observations at the best fit point for I and V
write(*,*) ’Phases at minumum from I: ’
do i=1,nFiles
write(*,*) ’JD: ’, JD(i), ’phase: ’, phasesI(i,iminI)
enddo
write(*,*) ’Phases at minumum from V: ’
do i=1,nFiles
write(*,*) ’JD: ’, JD(i), ’phase: ’, phasesV(i,iminV)
enddo
c
write the minumum points in the periodogram
write(*,*) ’Minimum in I at ’, pspecIT(iminI,1), ’days, chi^2: ’,
*
pspecIT(iminI,2)
write(*,*) ’Minimum in V at ’, pspecVT(iminV,1), ’days, chi^2: ’,
*
pspecVT(iminV,2)
c
and the statistical limit based on those points (for a 1st order fit)
if(iFito.eq.1)then
limI = pspecIT(iminI,2)+11.3/(real(nFiles) -3.0)
limV = pspecVT(iminV,2)+11.3/(real(nFiles) -3.0)
elseif(iFito.eq.2)then
limI = pspecIT(iminI,2)+15.1/(real(nFiles) -5.0)
limV = pspecVT(iminV,2)+15.1/(real(nFiles) -5.0)
endif
1
write(*,*) ’99% confidance limit on I:’, limI,
’99% confidance limit on V:’, limV
DEALLOCATE(pspecI, pspecIT)
DEALLOCATE(pspecV, pspecVT)
DEAllOCATE(phasesI, phasesV)
end
191
APPENDIX B. COMPUTER PROGRAMS WRITTEN
*******************************************************************************
C
Subroutine-ized (badly) by Colin for pbp.f (for pixel by pixel period searching)
C
(hacked for g95 compatibility by Colin)
C
C
MODIFIED AND UPDATED FOR EXECUTION IN UNIX BY GREGG WADE 1993.
C
C
INPUT IS FROM ’FOUFITDATA’. FORMAT MUST BE:
C
NUMBER OF DATA, PERIOD AND EPOCH ON LINE 1, THEN DATE..DATA..WEIGHT
C
FOR EACH ENTRY. OUTPUT FILE IS ’periods’.o
C
c
c
c
c
c
c
c
c
c
c
c
c
Takes: NMAX (number of points in the time series)(usualy N=NMAX),
P (a guss period, input is vestigial, used in a large loop),
FJDL (a referance julian date),
M (order of the function used to fit),
FJD(N) (the julian date of each point),
V(N) (the amplitude of each point),
WT(N) (the error on each point, 1 sigma) Note: this gets modified!,
pmin (the starting period of the search),
pmax (the ending period of the search),
ipnum (the number of points in the search)
pspec (the returned power spectrum) (no input only output)
phases (the phases for each observation at each ipnum period)
c
c
c
c
c
Changed the way the loop over phases is set up to avoid overflowing
Now we loop ipnum times and then calculate which period the current
loop should be based on pmin+(# loops)*(period change per point)
This should be safe up to ~1e9 points and
the limit of single float (real) precision in period
c
subroutine fsrch(NMAX,P,FJDL, M, FJD,V,WT,
pmin,pmax,ipnum,pspec,phases,iteration)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION Z(18,18)
DIMENSION Q(17)
DIMENSION RC(17),RS(17)
DIMENSION S(17),C(17)
DIMENSION OC(1000),PH(1000)
INTEGER x,ipmin,ipmax,istep,lp, a,b, M, count, iteration, ipnum
COMMON Z,S,C
real pspec(ipnum,2)
real*8 WT(NMAX), V(NMAX), FJD(NMAX)
double precision rpmin,rpmax,rstep,pmin,pmax
real phases(NMAX,ipnum)
C
DEFINING FREQUENTLY USED CONSTANTS
PI2=6.283185
1
C
C
C
count = 1
Q(1)=1.0
Number of data, period and epoch WERE read from first 3 elements in file.
c
92
C
C
C
WRITE (*,92)NMAX,P,FJDL
format(I5,2x,F12.5,2x,f20.6)
ENTERING ALL INPUT FOR HARMONIC ANALYSIS
22
199
N=0
FFN1=0.0
N=N+1
192
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c
996
201
8887
c
epha=(FJD(N)-FJDL)/P
if(epha.lt.0.0) epha=epha+int(epha)+1
write(*,996)FJD(N),epha,V(N),WT(N)
wt(N)=1/(wt(N)*wt(N))
FORMAT(4F16.5)
FFN1=FFN1+1.0
IF(WT(N).NE.0.0)GO TO 201
FFN1=FFN1-1.0
IF(N.LT.NMAX)GO TO 199
format(f10.2,f10.2,i10)
write(*,*) ’PMIN=’,pmin,’
PMAX=’,pmax,’
# periods:’,ipnum
BOCWT=SIGMA*SIGMA*(N-1)
3
C
C
c
M2=2*M
M21=M2+1
M22=M2+2
M23=M2+3
SET THE Z MATRIX EQUAL TO ZERO
DO I=1,M23
DO J=1,M23
Z(I,J)=0.0
enddo
enddo
FROM HERE EACH OBS IS DEALT WITH ONE AT A TIME
new hottness:
rstep = (pmax-pmin)/dble(ipnum)
write(*,*) pmin,pmax,rstep,ipnum
c
c
loop over the periods to be searched
do lp = 1,ipnum
scale steps exponentialy increasing from pmin to pmax
p = real(pmin*(pmax/pmin)**(dble(lp)/dble(ipnum)))
simplistic error trapping
if(p.gt.pmax) then
write(*,*) ’WARNING: periodogram exceeding max period’
write(*,*) p
endif
c
c
r1=0.0
y=0.0
bocwt=0.0
do a=1,18
do b=1,18
z(a,b)=0.0
end do
end do
loop over the observed data points
DO I=1,N
FINDING THE PHASE
EPPH=(FJD(I)-FJDL)/P
c
C
x=EPPH
EPPH=EPPH-FLOAT(X)
if (EPPH.lt.0) EPPH=EPPH+1.
EPPH=dabs(EPPH)
54
C
C
C
PH(I)=EPPH
IF(PH(I).GT.0.0)GO TO 54
PH(I)=PH(I)+1.0
FM=PH(I)*PI2
FINDING C1 AND S1 COEFF
Q(2)=DCOS(FM)
Q(3)=DSIN(FM)
FINDING C2 AND S2 TO CM AND SM COEFF
DO J=4,M2,2
J1=J-1
J2=J1-1
BY THE SIN & COS OF SUMS OF ANGLES
193
APPENDIX B. COMPUTER PROGRAMS WRITTEN
Q(J)=Q(J2)*Q(2)-Q(J1)*Q(3)
Q(J+1)=Q(J1)*Q(2)+Q(J2)*Q(3)
enddo
Q(M22)=V(I)
SETTING UP NORMAL EQUATIONS FOR LSTSQ SOLN
DO J=1,M22
DO K=J,M22
Z(J,K+1)=WT(I)*Q(J)*Q(K)+Z(J,K+1)
enddo
enddo
enddo
C
C
C
SOLN BY LSTSQ AND RELABELLING
CALL LSTSQ(M21,FFN1)
c
loop over the terms in the fit function
DO J=1,M
C(J)=Z(M22,2*J)
S(J)=Z(M22,2*J+1)
RC(J)=Z(M23,2*J)
RS(J)=Z(M23,2*J+1)
enddo
DATA, RESULTS & STATISTICS TABULATION
BOCWT=0.0
C
c
loop over the observed data points
DO I=1,N
Y=Z(M22,1)
FM=PI2*PH(I)
DO J=1,M
Y=Y+S(J)*DSIN(J*FM)+C(J)*DCOS(J*FM)
enddo
OC(I)=V(I)-Y
BOCWT=BOCWT+WT(I)*OC(I)*OC(I)
phases(I,count)=PH(I)
enddo
CALCULATION OF THE ERROR
NN=N-2*M-1
GNN=NN
R1=BOCWT/GNN
C
pspec(count,1) = p
pspec(count,2) = r1
count = count+1
C
EQUAL PHASE INCREMENT RESULTS
V0=Z(M22,1)
END do
end
C
C
C
LEAST SQUARES SUBROUTINE
C
SUBROUTINE LSTSQ(N,FNO)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION Z(18,18)
COMMON Z
CONTROL FROM MAIN PROGRAM TRANSFERRED TO HERE
M=N+1
M1=M+1
FN=N
DO7 I=1,M
L=I+1
C
1
2
C
COMPUTATION OF DIAGONAL MATRIX ELEMENTS
DO1 K=1,I
IF(I.LE.K)GO TO 2
Z(I,L)=Z(I,L)-Z(K,L)*Z(K,L)
IF(M.LE.I)GO TO 8
Z(I,L)=DSQRT(Z(I,L))
L1=L+1
DO4 J=L1,M1
COMPUTATION OF ELEMENTS TO RIGHT OF DIAGONAL
194
APPENDIX B. COMPUTER PROGRAMS WRITTEN
3
4
C
5
6
7
C
8
C
C
9
C
10
C
DO3 K=1,I
IF(I-K)4,4,3
Z(I,J)=Z(I,J)-Z(K,L)*Z(K,J)
Z(I,J)=Z(I,J)/Z(I,L)
Z(I,I)=1.0/Z(I,L)
DO6 J=1,I
IF(I.LE.J)GO TO 7
PP=0.0
L1=I-1
COMPUTATION OF ELEMENTS TO LEFT OF DIAGONAL
DO5 K=J,L1
PP=PP+Z(K,L)*Z(K,J)
Z(I,J)=-Z(I,I)*PP
CONTINUE
EVALUATION OF R1
Z(M1,M1)=0.675*DSQRT(Z(M,M1)/(FNO-FN))
DO10 I=1,N
Z(M,I)=0.0
PP=0.0
DO9 J=I,N
EVALUATION OF THE UNKNOWNS
Z(M,I)=Z(M,I)+Z(J,I)*Z(J,M1)
EVALUATION OF RECIPROCAL WEIGHTS
PP=PP+Z(J,I)*Z(J,I)
EVALUATION OF PROBABLE ERRORS OF UNKNOWNS
Z(M1,I)=Z(M1,M1)*DSQRT(PP)
RETURN CONTROL TO MAIN PROGRAM
RETURN
END
************************************************************************
subroutine interp(wl1, fl1, len1, wl2, fl2, len2, fl3)
C
assumes 1 is longer then 2 (or equal to)
integer len1, len2, start, jump
double precision wl1(len1), wl2(len2), fl1(len1), fl2(len2),
2
fl3(len1)
integer first, i,j
c
initialize to 0
do i=1,len1
fl3(i) = 0.0
enddo
c
advance 1 untill it overlaps with 2
start=1
do while(wl1(start).lt.wl2(1))
start=start+1
enddo
j=1
jump = 0
do i=start, len1
c
c
c
c
c
if we’re in range and 1 is increasing
if( (j.le.len2).and.(wl1(i).ge.wl1(i-1)) )then
advance to the next point in 2
do while( (wl2(j).lt.wl1(i)).and.(j.lt.len2) )
j=j+1
if there is a discontinuity we will want to remeber this
if(wl2(j).lt.wl2(j-1))then
jump = j
write(*,*) ’Warning: Discontinuity’
endif
now wl2(j-1) < wl1(i) and wl2(j) >= wl1(i)
enddo
if exact then use it
if(wl2(j).eq.wl1(i))then
fl3(i) = fl2(j)
195
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c
otherwise interpolate
else
fl3(i) = (wl1(i)-wl2(j-1))*(fl2(j)-fl2(j-1))/
(wl2(j)-wl2(j-1))+fl2(j-1)
endif
2
c
c
c
c
c
c
c
c
c
c
c
c
196
if we’re in range but 1 is decreasing
Note: This implememtation assumes that 2 and 1 have similar overlap in their orders.
The number of overlapping reigons in 1 and 2 must be the same.
The resulting overlap follows 1’s lambda values.
Any points in 2’s 1st order after 1 stopps overlapping will be ignored
(only the second order is used)
Any points in 2’s 2nd order berore 1 starts overlapping will be ignored
(only the 1st order is used)
Interpolation of end points is (almost) usless
Any points in 1’s 2nd order before 2’s 2nd order begins will exist
with (almost) junk values
Any points in 1’s 1st order after 2 stops will have values from 2’s second order
elseif( (j.le.len2).and.(wl1(i).lt.wl1(i-1)) )then
advance untill we decend in 2 too, unless we allready have then go to where we did decend
if(jump.eq.0)then
do while( (wl2(j).gt.wl2(j-1)).and.(j.lt.len2) )
j=j+1
enddo
else
j=jump
endif
c
c
advance to the next point in 2 straddeling this point in 1
do while( (wl2(j).lt.wl1(i)).and.(j.lt.len2) )
j=j+1
enddo
c
if exact then use it
if(wl2(j).eq.wl1(i))then
fl3(i) = fl2(j)
otherwise interpolate
else
fl3(i) = (wl1(i)-wl2(j-1))*(fl2(j)-fl2(j-1))/
(wl2(j)-wl2(j-1))+fl2(j-1)
endif
reset jump for the next overlap
jump = 0
c
2
c
c
if we’re out of range
else
if we’ve ran out of points use 0
fl3(i)=0.0
endif
c
enddo
end
APPENDIX B. COMPUTER PROGRAMS WRITTEN
B.3
197
Spectrum Fitting Through χ2 Minimization:
lma.f
This routine uses the Levenberg-Marquardt χ2 minimization method, presented in
Press et al. (1992), to fit a synthetic spectrum to an observed spectrum. Synthetic
spectra are calculated by ZEEMAN2. Free parameters in the fit are chemical abundances, v sin i, and microturbulence. The radial velocity of the star must be determined in advance, as must Teff , log g, and the magnetic properties of the star.
The program in broken into three files. The file lma.f performs the χ2 minimization
and calls all other files. The file rewriter.f formats the input for ZEEMAN2 and
writes it to the file ‘inzmodel.dat’ in the ‘data’ directory. The file z2.1v2mod1b-sub.f
contains the ZEEMAN2 synthesis code itself. ZEEMAN2 writes synthetic spectra
to a file where it is read back in by the ‘interp’ subroutine in lma.f. While this is
moderately inefficient it allows one to examine spectra as the routine operates, and
moreover it requires the minimum of modifications to ZEEMAN2.
Note that the ZEEMAN2 code itself has been omitted, as the author made only
minimal modifications to it for this thesis. In the source code of lma.f the ZEEMAN2
subroutine is named ‘ZeeModel’.
B.3.1
Levenberg-Marquardt χ2 Minimization
The lma.f file, containing the majority of the spectrum fitting routine.
C Compilation Requires z2.1v2mod1b-sub.f and rewriter.f
C Execution requires inlma.dat ./data/newatom.dat ./data/irwinpf.dat ./data/inzmodel.dat
C
C
C
C
C
CHECK LIST
Have you:
[ ] put the right file in observed.dat
[ ] input your initial estimate parameters
[ ] set the appropriate elements to be fitable
APPENDIX B. COMPUTER PROGRAMS WRITTEN
C
C
C
C
C
C
C
C
C
[
[
[
[
[
[
[
[
[
]
]
]
]
]
]
]
]
]
C
TO DO:
C
c
c
c
C
c
c
c
c
c
TO MAYBE DO:
Base continuum decision on whether the spectrum changes under spectrum synthesis
(% of deepest line?)
Auto fit for Vr (before everything else re: Gregg’s suggestion)
Check success criteria
Check numerical derivative lambda shift values
Make ’observed’ sigma values dynamic / more sensible
Read in success criteria, derivative shift, & sigma values from file (no linger hard coded)
IF you don’t care about the covariance matrix then there is 1 iteration more the necessary
(last derivatives and prettifying call)
198
set vsini to be fitable or fixed
set ma to the number of input parameters
set ndata to the length of observed.dat
set Itot in zeeman
checked the success criteria
checked the continuum cutoff level
checked the read format for observed.dat
backed up results.dat and inzmodel.dat
deleted any leftover w0* files
IMPLICIT NONE
integer ndata, ma, nca, fnCall, n, m
real chisq, alamda
integer stab
real chio, inichi, xlast, sucess, contcut
c
c
c
C
c
c
c
c
My attempt at fancy fortran 90 style dynamic arrays
REAL, DIMENSION(:), ALLOCATABLE :: x, y, sig, a
REAL, DIMENSION(:,:), ALLOCATABLE :: covar, alpha
integer, DIMENSION(:), ALLOCATABLE :: ia
these three are passed in and out of mrqmin to preserve values
(could use SAVE if they were not dynamic)
REAL, DIMENSION(:), ALLOCATABLE :: atry, beta, da
Commonly Changed Values
sucess: Fractional change in chi^2 less then which we say the fit is ok
(or not likely to get significantly better) Must ocur 2 times in a row (by default) to terminate
sucess = 0.005
contcut: The continuum level cutoff. Used to reject points that are just continuum from the fit
can be set to >> 1 and (virtualy) the entire spectrum in the synthisized range will be fit
contcut = 10.0
c
c
c
c
c
c
c
c
ndata = # obs. data pts. ma = # input param. nca = fitting matrix size (>=ma),
fcCall = function call counter, n,m=counters, stab = ’stability’ check counter
chisq = chi squared, alamda = shift in lambda / flag, chio = old chisq,
inichi = initial chisq, xlast = last x (checks for overlap in observed.dat)
x,y = oberved data points (wavelength, normalized intensity)(ndata long)
sig = standard deveation on y (presently just made up) a = input parameters
ia = input parameters to fit (0=no/1=yes), covar = covariance matrix (output ignored),
alpha = augmented (w lambda) Hessian matrix ~(d^2(chi^2)/(da_i)(da_j))
c
c
c
c
c
c
Definition of parameters in a(n) and ia(n):
n=1 => radial velocity (m/s)
n=2 => vsini (cm/s)
n=3 => micro turbulance (cm/s)
n=i (i=4,6,8,...) => atomic number
n=i+1 abundance for atom # in n=i
c
open(15, file="observed.dat", status=’unknown’)
open(16, file="results.dat", status=’unknown’)
open(17, file="inlma.dat", status=’unknown’)
open(19, file="dump.dat", status=’unknown’)
C
Read in the input parameters and dimensions file.
read(17,219)
read(17,210) ndata, ma
APPENDIX B. COMPUTER PROGRAMS WRITTEN
ALLOCATE(a(1:ma))
ALLOCATE(ia(1:ma))
And the three storage arrays for mrqmin
ALLOCATE(atry(1:ma))
ALLOCATE(beta(1:ma))
ALLOCATE(da(1:ma))
c
read(17,219)
read(17,211) a(1), ia(1)
read(17,219)
read(17,211) a(2), ia(2)
read(17,219)
read(17,211) a(3), ia(3)
read(17,219)
do n=4,ma,2
read(17,212) a(n), a(n+1), ia(n+1)
never fit the atomic #
ia(n) = 0
enddo
c
close(17)
c
set size of fitting matrix, >= ma
nca = ma
210
211
212
219
format(I7,1X,I3)
format(E11.2,1X,I2)
format(F3.0,6X,F9.6,1X,I2)
format(1X)
write(*,*) ’Initial Param:’
write(16,*) ’Initial Param:’
write(*,*) a
write(16,*) a
write(*,*) ’Free Param:’
write(16,*) ’Free Param:’
write(*,*) ia
write(16,*) ia
c
c
c
There are 10 arrays (5 here 2+3 above)
be kind and deallocate them all (though it should happend automaticaly)
ALLOCATE(x(1:ndata))
ALLOCATE(y(1:ndata))
ALLOCATE(sig(1:ndata))
ALLOCATE(covar(1:nca,1:nca))
ALLOCATE(alpha(1:nca,1:nca))
... that was too easy...
c
Read in observed data (assumes wavelength in angstroms, no overlap, normalized to 1)
xlast = 0.0
do n=1,ndata
read(15,200) x(n), y(n)
200
format(F10.4,1X,E11.4)
c 200
format(f21.16,1x,f10.7)
c 200
format(3X,E13.7,3x,E13.7)
c 200
format(f11.6,1x,f8.6)
c
convert nm to angstroms
x(n)=x(n)*10.0
c
simple error trapping
if(x(n).lt.xlast)then
write(*,*)"ERROR! overlap in the observed data"
write(16,*)"ERROR! overlap in the observed data"
endif
xlast = x(n)
enddo
close(15)
c
Apply the doppler shift correction to the observed data
do n=1,ndata
x(n) = x(n)- x(n)*a(1)/2.99792458E8
199
APPENDIX B. COMPUTER PROGRAMS WRITTEN
enddo
c
Estimate sigma (the uncertainty on y) (to improve...)
do n=1,ndata
sig(n) = 1.000
enddo
c
Initialize everything else to 0
do n=1,nca
do m=1,nca
covar(n,m) = 0.0
alpha(n,m) = 0.0
enddo
enddo
chisq = 0.0
alamda = -1.0
stab = 0
fnCall = 0
chio = 0.0
C
Run the initialization call of mrqmin
write(*,123) 1
write(16,123) 1
**************************************************************
call mrqmin(x,y,sig,ndata,a,ia,ma,covar,alpha,nca,chisq,
1
alamda,fnCall,contcut, atry,beta,da)
**************************************************************
inichi = chisq
write(*,124) chisq, chisq-chio
write(16,124) chisq, chisq-chio
write(*,*) a
write(16,*) a
C
Iterate to fit. We must have less then a ’sucess’ fractional change 2 times in a row
n=1
do while ( (abs(chisq-chio)/chio.gt.sucess).or.(stab.lt.2) )
chio = chisq
write(*,123) n+2
write(16,123) n+2
format(’Iteration: ’, I4)
123
**************************************************************
call mrqmin(x,y,sig,ndata,a,ia,ma,covar,alpha,nca,chisq,
1
alamda,fnCall,contcut, atry,beta,da)
**************************************************************
124
1
write(*,124) chisq, chisq-chio, (chisq-chio)/chio
write(16,124) chisq, chisq-chio, (chisq-chio)/chio
format(’chi squared: ’,F16.6,’ change: ’,F14.6,
’ fractional change: ’ F10.6)
write(*,*) a
write(16,*) a
if(abs(chisq-chio).le.sucess) then
stab = stab + 1
else
stab = 0
endif
n=n+1
enddo
c
Run the finalization call of mrqmin and output the results
alamda = 0.0
write(*,*) ’Finalizing’
200
201
APPENDIX B. COMPUTER PROGRAMS WRITTEN
write(16,*) ’Finalizing’
**************************************************************
call mrqmin(x,y,sig,ndata,a,ia,ma,covar,alpha,nca,chisq,
1
alamda,fnCall,contcut, atry,beta,da)
**************************************************************
write(*,*)
write(*,*)
write(*,*)
write(*,*)
’final fit param’
a
’final chi squared and change’
chisq, chisq-inichi
write(16,*) ’covariance matrix’
do n=1,nca
write(16,*) covar(n,1:nca)
enddo
write(16,*)
write(16,*)
write(16,*)
write(16,*)
’final fit param’
a
’final chi squared and change’
chisq, chisq-inichi
DEALLOCATE(x)
DEALLOCATE(y)
DEALLOCATE(sig)
DEALLOCATE(a)
DEALLOCATE(ia)
DEALLOCATE(covar)
DEALLOCATE(alpha)
DEALLOCATE(atry)
DEALLOCATE(beta)
DEALLOCATE(da)
c
thats 10 down
c
close(16)
close(19)
END
********************************************************************
********************************************************************
********************************************************************
C Levenberg-Marquard non-linear chi squared minimization routine
C based on the algorithm and code from Numerical Recipes (J-M’s copy)
C
c
The major fitting function. Has initilization (alamda < 0), iteration (alamda = 0),
and finalization (alamda = 0) modes
C In Detail:
c
Initialization: (initializes, runs zeeman, gets 1st chi^2 and derivatives)
c
sets mfit and alambda
c
calls mrqcof getting alpha beta and chisq (with inital parameters)
c
sets ochisq to chisq and atry to a
c
goes straight to the first real iteration
c
Iteration: (uses derivatives to guss next a, runs zeeman with next a, gets chi^2 and derivatives,
c
keeps them if good otherwise reverts to the last iteration)
c
sets covar to alpha (with an augmented diagonal) and da to beta
c
runs gaussj (gauss jordain elimination to solve for the next step)
c
sets the next attempt values atry
c
calls mrqcof getting alpha beta and chisq (with atry parameters)
c
if it’s sucessfull shrink alamda, set alpha to covar, beta to da, a to atry, and ochisq to chisq
c
if not grow alamda
c
Finalization: (calls covsrt to get the covariance matrix and mrqcof to get a final spectrum)
c
still sets covar to alpha (with an augmented diagonal) and da to beta
c
still runs gaussj (gauss jordain elimination to solve for the next step)
c
calls covsrt to reorganize covar into the covariance matrix
c
calls mrqcof (using atry) to get the final spectrum (no derivatives this time)
C
x(ndata), y(ndata), sig = input data with 1 sigma uncertainties.
ndata long.
APPENDIX B. COMPUTER PROGRAMS WRITTEN
C
C
C
C
C
C
C
a(ma) initial input parameters. ma long. ia(ma) parameters to fit (1=fit, 0=leave)
covar(nca,nca) = covariance matrix, returned also used as a workspace for alpha.
alpha(nca,nca) = modified curvature | Hessian matrix, eq 15.5.11, returned.
nca size of covar and alpha, >= ma, must be given.
chisq = chi squared. alamda = new ’lamda’, adjust the importance
of the diagonal elements in " a’ ", alters our next pt guss. Must be < 0 on first call
Must be = 0 for final evaluation, otherwise use what was returned.
1
SUBROUTINE mrqmin(x,y,sig,ndata,a,ia,ma,covar,alpha,nca,chisq,
alamda,fnCall,contcut, atry,beta,da)
IMPLICIT NONE
c
c
c
c
c
c
c
c
c
C
C
c
integer ma,nca,ndata,ia(ma),fnCall
ia = things to fit for (!= 0) ndata=size of data to fit,
ma = # param, nca (>= ma), fnCall mostly just perserved for fun (counts zeeman calls)
real alamda,chisq, a(ma), alpha(nca,nca), covar(nca,nca),
1
sig(ndata), x(ndata), y(ndata), contcut
alamda = change in lamda (or flag) chisq=obvious,
a = input param, alpha = eq15.5.11, covar = covariance matrix,
sig = uncertainties, x,y = data points
integer j,k,l,m,mfit
mfit = number of parameters were fitting <= ma
real ochisq, atry(ma), beta(ma), da(ma)
beta = eq 15.5.8, atry = temporary a, da = temporary beta
SAVE ochisq, atry, beta, da, mfit (numerical recipies used this with fixed length arrays)
SAVE ochisq, mfit
dummy, dummy2: just used to hold the output of the funcs subroutine
real dummy(ndata), dummy2(ndata,ma)
Initialization - use alamda < 0
normaly alamda is used to return the suggested change to the input parameters a
if(alamda .lt. 0.0)then
mfit = 0.0
do j=1, ma
if (ia(j).ne.0) mfit=mfit+1
enddo
set initial alamda value
alamda = 0.001
********************************
call mrqcof(x,y,sig,ndata,a,ia,ma,alpha,beta,nca,chisq,
1
fnCall,alamda,contcut)
********************************
ochisq = chisq
do j=1,ma
atry(j)=a(j)
enddo
write(*,*) ’Iteration: 2’
write(16,*) ’Iteration: 2’
endif
C
c
c
set up the covar matrix as the altered linearized fitting matrix
j=0
do l=1,ma
if(ia(l).ne.0) then
j=j+1
k=0
do m=1,ma
if(ia(m).ne.0)then
k=k+1
set the simple/standard elements
covar(j,k)=alpha(j,k)
endif
enddo
set the (augmented) diagonal elements
covar(j,j)=alpha(j,j)*(1.0+alamda)
da(j)=beta(j)
202
APPENDIX B. COMPUTER PROGRAMS WRITTEN
endif
enddo
********************************
call gaussj(covar,mfit,nca,da,1,1)
c
find the matix solution (via Gauss Jordan Elimination)
********************************
C
once converged evaluate once more and return the results
if(alamda.eq.0.0)then
*********************************
call covsrt(covar,nca,ma,ia,mfit)
*********************************
c
and make a pretty output
*********************************
call funcs(x, a, ia, dummy, dummy2, ma, ndata,fnCall,alamda)
*********************************
return
endif
C
did this iteration improve chi squared? ...
j=0
do l=1,ma
if(ia(l).ne.0) then
j=j+1
atry(l)=a(l)+da(j)
endif
enddo
*******************************
call mrqcof(x,y,sig,ndata,atry,ia,ma,covar,da,nca,chisq,
1
fnCall,alamda,contcut)
*******************************
C
c
c
c
c
c
if so then accept the improved solution
if(chisq.lt.ochisq)then
tighten alamda (default: multiply by 0.1)
alamda=0.1*alamda
ochisq=chisq
set alpha to the improved (covar) values
j=0
do l=1,ma
if(ia(l).ne.0)then
j=j+1
k=0
do m=1,ma
if(ia(l).ne.0)then
k=k+1
alpha(j,k)=covar(j,k)
endif
enddo
accept the da (beta) and atry (a) values
beta(j) = da(j)
a(l) = atry(l)
endif
enddo
if not then increase alamda and leave the old best values
else
alamda = 10.0*alamda
chisq=ochisq
endif
return
END
********************************************************************
203
APPENDIX B. COMPUTER PROGRAMS WRITTEN
********************************************************************
********************************************************************
C
C
C
C
C
mrqcof Evaluates the liniarized fittimg matrix alpha and the vector beta (eq 15.5.8)
and calculates chi squared. Takes: x, y, sig, ndata, a, ia, ma, nalp, fnCall, alamda
Returns alpha beta, chisq, fnCall (just passed from funcs)
Same variable names used as in mrqmin, except nalp = nca
and in one instance atry = a, covar = alpha, da = beta
C More explicitly it:
c
initialized alpha, beta and chisq to 0
c
calls funcs - the zeeman interface function
c
for calculated points != 0 and observed points > conlvl
c
add to the sum over all points for alpha beta and chisq
1
SUBROUTINE mrqcof(x,y,sig,ndata,a,ia,ma,alpha,beta,nalp,
chisq,fnCall,alamda,contcut)
IMPLICIT NONE
c
c
c
c
c
c
c
c
integer ma, nalp, ndata, ia(ma),fnCall
ia = things to fit for (!= 0), ndata=size of data to fit,
ma = # param, nalp (>= ma), fnCall mostly just perserved for fun (counts zeeman calls)
real chisq, a(ma), alpha(nalp,nalp), beta(ma), sig(ndata),
1
x(ndata), y(ndata), alamda, contcut
chisq=obvious, a = input param, alpha = eq15.5.11 ,beta = eq 15.5.8
sig = uncertainties, x,y = data points, alamda = here only flag for finalization (0.0=end)
integer mfit, i, j, k, l, m
mfit = number of parameters were fitting <= ma
real dy, sig2i, wt, ymod(ndata), dyda(ndata,ma), conlvl
dyda was MMAX long but MMAX has to be ma for this to work so I tried to make life eaiser
set mfit
mfit = 0
do j=1,ma
if(ia(j).ne.0) mfit = mfit+1
enddo
initialize (1/2 of symmetric) alpha and beta
do j=1,mfit
do k=1,j
alpha(j,k)=0.0
enddo
beta(j)=0.0
enddo
chisq = 0.0
*****************************************
call funcs(x, a, ia, ymod, dyda, ma, ndata,fnCall,alamda)
C
zeeman calling function.
*****************************************
cccccccccccccccccccccccccccccccccccccccccccccccccccc
open(19, file="dump.dat", status=’unknown’)
cccccccccccccccccccccccccccccccccccccccccccccccccccc
c
Summation loop over all data
do i=1,ndata
c
Only use points for which we have a calculated y
if( (ymod(i).ne.0.0).and.(y(i).lt.contcut) )then
c
if(ymod(i).ne.0.0)then
c
c
C
c
set sig2i (sigma squared inverse) to 1/sigma^2
sig2i = 1.0/(sig(i)*sig(i))
find the diff between obs and calc
dy=y(i)-ymod(i)
calculate the new alpha and beta
j=0
do l=1,ma
if(ia(l).ne.0)then
j=j+1
set the ’weighting’ of this point: (1/sigma^2)*(dy/da_j)
wt=dyda(i,l)*sig2i
204
APPENDIX B. COMPUTER PROGRAMS WRITTEN
k=0
do m=1,l
if(ia(m).ne.0)then
k=k+1
c
alpha = sum((1/sigma^2)*(dy/da_j)*(dy/da_k)) (eq 15.5.11)
alpha(j,k)=alpha(j,k)+wt*dyda(i,m)
endif
enddo
c
beta_j = -1/2*(dChi^2/da_j)^2 = sum((y_o - y)*(1/sigma^2)*(dy/da_j)) (eq 15.5.8)
beta(j) = beta(j)+dy*wt
endif
enddo
c
calcuate chi squared
chisq=chisq+dy*dy*sig2i
ccccccccccccccccccccccccccccccccccccccccccc
write(19,*) x(i), y(i), ymod(i)
ccccccccccccccccccccccccccccccccccccccccccc
endif
enddo
c
Fill in the other side of the symmetric alpha
do j=2,mfit
do k=1,j-1
alpha(k,j)=alpha(j,k)
enddo
enddo
ccccccccccccccccccccccccccccccccccccccc
close(19)
ccccccccccccccccccccccccccccccccccccccc
return
END
********************************************************************
********************************************************************
********************************************************************
C gaussj Performs Gauss Jordan Elimination with ’full pivoting’
C
C
C
SUBROUTINE gaussj(a,n,np,b,m,mp)
Takes input matrices a(1:n,1:n) and b(1:n,1:m)
with array physical sizes np X np and np X mp.
Returns the solution arrays a(n,n) and b(n,n)
IMPLICIT NONE
c
c
c
integer m, mp, n, np, NMAX
real a(np,np), b(np,mp)
PARAMETER(NMAX=20)
integer i, icol, irow, j, k, l, ll, indxc(NMAX), indxr(NMAX),
1
ipiv(NMAX)
integer i, icol, irow, j, k, l, ll, indxc(n), indxr(n),
1
ipiv(n)
C
ipiv indxr and indxc are used for pivoting bookkeeping
real big, dum, pivinv
c
initialize ipiv
do j=1,n
ipiv(j) = 0
enddo
C
The main loop over all columns to be reduced
do i=1,n
Search for a good pivor element, looking over all rows (j) and colums (k) of a
big = 0.0
do j=1,n
if(ipiv(j).ne.1)then
do k=1,n
C
205
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c
c
if(ipiv(k).eq.0)then
if we have the bigest number yet the it’s a good candidate, keep it
if(abs(a(j,k)).ge.big)then
big = abs(a(j,k))
irow = j
icol = k
endif
else if (ipiv(k).gt.1)then
write(*,*) ’singular matrix in gaussj [1]’
write(16,*) ’singular matrix in gaussj [1]’
error traping. bad input?
endif
enddo
endif
enddo
ipiv(icol)=ipiv(icol)+1
c
c
c
c
c
c
c
c
c
c
c
c
c
Now we have a good pivot element we can interchange rows, if needed,
to put the pivot element on the diagonal. This is done by relabeling.
indxc(i), the column of the ith pivot element, is the ith column reduced,
while indxr(i) is the row in which that pivot element was origonaly located.
If indxc(i) != indxr(i) then there is an implied column interchange.
Thus the solution b’s will end up in the correct order but the inverse
matrix will be scrabled by columns
if (irow.ne.icol) then
do l=1,n
dum = a(irow,l)
a(irow,l) = a(icol,l)
a(icol,l) = dum
enddo
do l=1,m
dum=b(irow,l)
b(irow,l) = b(icol,l)
b(icol,l) = dum
enddo
endif
Now we devide the pivot row by the pivot element (point at irow, icol) renormalizing
indxr(i) = irow
indxc(i) = icol
if(a(icol,icol).eq.0.0) then
error traping. bad input?
write(*,*) ’singular matrix in gaussj [2]’
write(16,*) ’singular matrix in gaussj [2]’
endif
pivinv = 1.0/a(icol,icol)
a(icol,icol)=1.0
do l=1,n
a(icol,l)=a(icol,l)*pivinv
enddo
do l=1,m
b(icol,l)=b(icol,l)*pivinv
enddo
Now reduce the rows, except for the pivot row
do ll=1,n
if(ll.ne.icol)then
dum=a(ll,icol)
a(ll,icol)=0.0
do l=1,n
a(ll,l)=a(ll,l)-a(icol,l)*dum
enddo
do l=1,m
b(ll,l)=b(ll,l)-b(icol,l)*dum
enddo
endif
enddo
enddo
end of the major loop over columns
now we just need to unscrable the solution
interchange pairs of columns in the reverse order
do l=n,1,-1
if(indxr(l).ne.indxc(l))then
206
APPENDIX B. COMPUTER PROGRAMS WRITTEN
do k=1,n
dum=a(k,indxr(l))
a(k,indxr(l))=a(k,indxc(l))
a(k,indxc(l))=dum
enddo
endif
enddo
return
END
********************************************************************
********************************************************************
********************************************************************
C
C
C
C
covsrt Propogates the proper order of entries in back into the full ma X ma
covariance matrix ’covar’. Only realy necessary to enshure proper output.
not important to me as I don’t realy care about the covariance of the input
parameters for zeeman (at least not now).
SUBROUTINE covsrt(covar, npc, ma, ia, mfit)
C
C
C
covar = the covariance matrix (output) npc X npc large, npc = size of covar,
ma = size of ia (and a), ia = which parameters are free (1|0),
mfit = # of param to be fit
IMPLICIT NONE
integer ma, mfit, npc, ia(ma)
real covar(npc,npc)
integer i, j, k
real swap
c
c
c
set unused elements of covar to 0
do i = mfit+1,ma
do j=1,i
covar(i,j)=0
covar(j,i)=0
enddo
enddo
interchange the non-zero elements of covar to give the right order
k=mfit
do j = ma,1,-1
if(ia(j).ne.0)then
do i=1,ma
swap = covar(i,k)
covar(i,k)=covar(i,j)
covar(i,j)=swap
enddo
do i=1, ma
swap = covar(k,j)
covar(k,i)=covar(j,i)
covar(j,i)=swap
enddo
k=k-1
endif
enddo
c’est tout
return
END
****************************************************************************
****************************************************************************
****************************************************************************
c
THE (zeeman) FUNCTION SUBROUTINE: funcs(x, a, ymod, dyda, ma, ndata)
207
APPENDIX B. COMPUTER PROGRAMS WRITTEN
****************************************************************************
****************************************************************************
****************************************************************************
c
c
c
c
c
c
Definition of parameters in a(n) and ia(n):
n=1 => radial velocity (m/s)
n=2 => vsini (cm/s)
n=3 => micro turbulance (cm/s)
n=i (i=4,6,8,...) => atomic number
n=i+1 abundance for atom # in n=i
C
C
SUBROUTINE funcs(x,a,ia,y,dyda,na,ndata,fnCall,alamda)
takes arrays: x, a, ia, real: alamda, and ints: na, ndata, fnCall
returns arrays y (for the corrisponding x), and dyda (for the corrisponding y and a)
IMPLICIT NONE
integer na, ndata
real x(ndata), y(ndata), a(na), dyda(ndata,na), alamda
integer ia(na)
here alamda = only a flag for finalization (0.0 = end)
c
integer nWindows, fnCall, chkdyda
real xt(ndata), yt(ndata), at(na), epsilon, epsilonE, epsilonV
integer i,j,k,l
nWindows = 0
c
c
the small _fractional_ shift in a parameter for it’s numerical derivative
may need to be changed depending on the parametere’s size and stability (this assumes ~1)
epsilon = 0.02
elements specific epsilon (absolute)
epsilonE = 0.05
velocities specific epsion (absolute)
epsilonV = 0.5E5
c
c
c
c
First check if inputs are in rage (-1..-13 dex in abundance)
If it’s not then put it back in range and complaine.
do i=5,na,2
if(a(i).gt.-1.0)then
write(*,231) i, a(i)
write(16,231) i, a(i)
a(i) = -1.0
else if(a(i).lt.-13.0)then
write(*,232) i, a(i)
write(16,232) i, a(i)
a(i) = -13.0
endif
enddo
231 format(’ERROR: parameter ’,I3,’ too large: ’,F9.4,
2
’ setting to -1.0’)
232 format(’ERROR: parameter ’,I3,’ too small: ’,F9.4,
2
’ setting to -13.0’)
c
233
c
And make sure microturbulance is >= 0
if( (ia(3).ne.0) .and. (a(3).lt.0.0) )then
write(*,233) a(3)
write(16,233) a(3)
a(3) = 0.0
endif
format(’ERROR: parameter 3 too small: ’,E10.3,’ setting to 0.0’)
Setup, Run, and Read the results of Zeeman
****************************************************
call rewriter(a ,ia, na, nWindows)
c
write this iteration into zeeman’s input file
c
currently an external file: rewriter.f
****************************************************
****************************************************
208
APPENDIX B. COMPUTER PROGRAMS WRITTEN
call ZeeModel()
c
an external file, currently: z2.1v2mod1b-sub.f
****************************************************
****************************************************
call interp(x, y, ndata, nWindows)
c
read zeeman’s output and interpolate linearly
****************************************************
226
c
c
c
c
fnCall = fnCall+1
write(*,226) fnCall
write(*,*)
write(16,226) fnCall
format(’Function Calls:’I4)
Now we have the calculated y vales we need the derivatives
(done numericaly) (if we’re not on the finalization call)
if(alamda.ne.0.0)then
first back up the input parameters
do i=1,na
at(i) = a(i)
enddo
then adjust each parameter in turn to find the change
do i=1,na
if(ia(i).ne.0)then
if(i.le.3) then
epsilon = epsilonV
else
epsilon = epsilonE
endif
c
c
at(i) = at(i)+epsilon*a(i)
at(i) = at(i)+epsilon
Now we need to evalate the function again
****************************************************
call rewriter(at ,ia, na, nWindows)
c
write this iteration into zeeman’s input file
c
currently an external file: rewriter.f
****************************************************
****************************************************
call ZeeModel()
c
an external file, currently: z2.1v2mod1b-sub.f
****************************************************
****************************************************
call interp(x, yt, ndata, nWindows)
c
read zeeman’s output and interpolate linearly
****************************************************
fnCall = fnCall+1
write(*,226) fnCall
write(*,*)
write(16,226) fnCall
c
chkdyda=0
do j=1,ndata
dyda(j,i) = (yt(j)-y(j))/(epsilon*a(i))
dyda(j,i) = (yt(j)-y(j))/(epsilon)
if( abs(dyda(j,i)).gt.0.0)then
chkdyda=1
endif
enddo
c
c
Simple error traping. If dyda has a row of 0s then there will be a error
(devide by 0) in the gauss jordan elimination subroutine. (better to catch it now)
209
APPENDIX B. COMPUTER PROGRAMS WRITTEN
if(chkdyda.eq.0)then
write(*,230) i
write(16,230) i
endif
format("Error: derivative of chi^2 wrt a = 0 for paramete
230
*r: ", I3)
at(i) = a(i)
endif
enddo
endif
return
END
****************************************************************************
****************************************************************************
****************************************************************************
c
THE interp SUB-SUBROUTINE
c
reads and interpolates zeeman output files. Given a set of x values (obervations)
c
the associated y values are calculated useing linear interpolation
C
C
C
C
C
C
C
Currently this assumes that the observed data is longer then the calculated (in wavelength)
Cannot currently deal with overlap in the observed spectrum
(asumes each line is at a > lambda then the last)
When there is overlap in the calculated windows it takes the second
(overwrites the 1st in the overlap reigon)
For points at observed wavelengths (x) where there is no calculated y value 1.0 is returned
(ie. the results are padded out to the observed spectrum length with 1.0s)
SUBROUTINE interp(x, y, ndata, nWindows)
IMPLICIT NONE
c
c
c
c
c
integer ndata, nWindows
real x(ndata), y(ndata)
x, y = observed values
the number of lines in one synthetic window
assume 15 A window length with a resolution of 0.01 A
this gives 1500 (+1) lines (ignore the last (the +1) line)
integer ncalc
PARAMETER(ncalc = 1501)
real xt(ncalc), yt(ncalc)
xt, yt = calculated x and y
integer i,j,k,l
character(5) fn1(40)
data fn1 /’./w01’,’./w02’,’./w03’,’./w04’,’./w05’,’./w06’,
1
’./w07’,’./w08’,’./w09’,
2
’./w10’,’./w11’,’./w12’,’./w13’,’./w14’,’./w15’,
1
’./w16’,’./w17’,’./w18’,’./w19’,
2
’./w20’,’./w21’,’./w22’,’./w23’,’./w24’,’./w25’,
1
’./w26’,’./w27’,’./w28’,’./w29’,
2
’./w30’,’./w31’,’./w32’,’./w33’,’./w34’,’./w35’,
1
’./w36’,’./w37’,’./w38’,’./w39’,’./w40’/
c
initialize y to 0.0
do i=1,ndata
y(i) = 0.0
enddo
c
first loop over all Zeeman’s output files
do i=1,nWindows
open(25,FILE=fn1(i)//’p01’,STATUS=’OLD’)
read this output file
c
210
APPENDIX B. COMPUTER PROGRAMS WRITTEN
do j=1,ncalc
read(25,225) xt(j), yt(j)
enddo
and close it before I forget
close(25)
c
c
c
c
now we need to interpolate between calculated points to get the
appropreate value for the observed x point
(done in a somewhat simplistic & inflexible fashion) (I should improve this!)
k=0
l=0
so loop over all the x’s we want a y for
do j=1,ndata
find the first appropreate calculated point (if we haven’t yet)
if( (x(j).gt.xt(1)).and.(k.eq.0) )then
k=1
(simiple linear interpolation)
y(j) = (yt(2)-yt(1) )*(x(j)-xt(1) )
/(xt(2)-xt(1) )+yt(1)
c
c
c
1
c
if we have previously found the first point and we haven’t run out of cacluated points
elseif( (k.gt.0).and.(l.le.(ncalc-1)) )then
scroll along to the next appropreate caclulated point (or stop when we run out)
do while( (xt(l).lt.x(j)).and.(l.le.(ncalc-1)) )
l=l+1
enddo
c
c
if we haven’t run out then use it
if(l.le.(ncalc-1))then
k=k+1
y(j) = (yt(l)-yt(l-1) )*(x(j)-xt(l-1) )
/(xt(l)-xt(l-1) )+yt(l-1)
endif
endif
1
enddo
enddo
225
format(f10.4, f20.10)
return
END
B.3.2
ZEEMAN2 Input and Output: rewriter.f
The rewriter.f file, containing the input formatting for ZEEMAN2.
SUBROUTINE rewriter(a ,ia, na, nWindows)
IMPLICIT NONE
C edits the inzmodel.dat file
C available parameters for editing: vsini, abundances (elements must exist allredy)
c
integer MaxWindows
the maximum number of windows allowed in inzemodle.dat
PARAMETER(MaxWindows = 40)
integer na, nWindows
real a(na)
integer ia(na)
211
APPENDIX B. COMPUTER PROGRAMS WRITTEN
integer i, j, k, nLines(MaxWindows), nEle
real vsini, vmic, abun(103,28)
character(26) comment
c
unused but read in
double precision WL0(MaxWindows), ALINLS(103,14,40),
1
DPWL(103,40),rlland(103,40),ruland(103,40),FW,renorm,PHASE,
2
RAID, tau0l(103), tau0u(103), delta_abun(103), frh,frhe,
3
AMASS(100),T(100),RNE(100),RNATK(100),RHO(100),
4
BETAD,BPDD,ADD,BQ,BOCT,razang
integer KPL,NPHAS,NOBS,isflag(103,40,4), MU
OPEN(15,FILE=’./data/inzmodel.dat’,STATUS=’OLD’)
C PART 1: READ IN THE FILE
READ(15,115) nWindows
FORMAT(I4)
the line list
do i=1, nWindows
115
c
READ(15,120) nLines(i),WL0(i)
FORMAT(I4,F10.3)
120
1
130
do j=1, nLines(i)
READ(15,130)(ALINLS(j,K,i),K=1,2),DPWL(j,i),
(ALINLS(j,K,i),K=4,14),rlland(j,i),ruland(j,i)
FORMAT(2F3.0,F10.4,6F4.1,F7.3,F10.3,3E10.3,2f6.3)
enddo
enddo
c
151
c
153
c
c
c
c
# of phases, # observatiosn??, instrumental profile, and renormalization
READ(15,151) NPHAS,NOBS,FW,renorm
FORMAT(2I4,2F10.0)
do i=1, nWindows
DO j=1,NPHAS
phase and stokes flags for I, V, Q, U
READ(15,153) PHASE,(isflag(j,i,K),K=1,4)
FORMAT(F10.3,4I4)
ENDDO
enddo
150
Read in ’RAID’ (inclination angle of rotation axis to line of sight
allways 90 so far),
vsini, microturbulance, and KPL (a flag for ’which intensity profile to plot’,
allways -1) seems to be vestigial
READ(15,150) RAID,vsini,vmic,KPL
FORMAT(F10.0,24X,2E10.1,I4)
160
Read in BETAD (angle between magnetic and rotation axies (usualy 90)),
BTDD (dipole mangetic field strenght), ADD ’decentering parameter’ for the dipole
BQ (quadrupole field strength), BOCT (octupole strength)(’co-linar with the rest),
razang (’position angle of rotation axis projected on the sky’)
READ(15,160) BETAD,BPDD,ADD,BQ,BOCT,razang
FORMAT(6F10.0)
170
uncertaint here here, except for the 1st entry is the number of elements
READ(15,170) nEle
format(i4)
180
Read In The Elements
do i=1,nEle
in abun: 1 = element, 2 = ???, 3 = abundance!
READ(15,180) (abun(i,J),J=1,9),tau0l(i),tau0u(i),delta_abun(i)
FORMAT(F2.0,11F8.0)
101
enddo
read the headder for the modle atmosphere
MU = # layers, comment = the original file
READ(15,101) MU,frh,frhe, comment
FORMAT(I2,F9.3,f13.3,A25)
c
c
c
c
c
C
c
c
c
212
APPENDIX B. COMPUTER PROGRAMS WRITTEN
C
The Model Atmosphere
do i=1,MU
READ(15,110) AMASS(i),T(i),RNE(i),RNATK(i),RHO(i)
FORMAT(E16.9,1X,F9.0,1X,3(E12.6,1X))
enddo
110
C PART 2: EDIT VAULES
c
x only if the parameter is set to be fittable right now x
c
x otherwise we leave that line alone x
c
write all parameters with values in ’inlma.dat’
c
c
Change the vsini
if(ia(2) .ne. 0)then
vsini = a(2)
endif
Change the microturbulance
if(ia(3) .ne. 0)then
vmic = a(3)
endif
Change the element abundance
do i=1,nEle
do j=4,na-1,2
if( (ia(j+1).ne.0).and.(a(j).eq.abun(i,1)) )then
if( a(j).eq.abun(i,1) )then
abun(i,3) = a(j+1)
endif
enddo
enddo
c
c
c
c
c
c
C PART 3: WRITE THE FILE
REWIND(15)
WRITE(15,115) nWindows
do i=1, nWindows
WRITE(15,121) nLines(i),WL0(i)
FORMAT(I4,F8.2)
121
c
The Line List
do j=1, nLines(i)
WRITE(15,131) int(ALINLS(j,1,i)), int(ALINLS(j,2,i)),
DPWL(j,i),(ALINLS(j,K,i),K=4,14),rlland(j,i),ruland(j,i)
FORMAT(2I3,F10.4,6F4.1,F7.3,F10.3,3E10.3,2f6.3)
1
131
enddo
enddo
152
c
c
154
WRITE(15,152) NPHAS,NOBS,FW,renorm
FORMAT(I3,I4,F11.5, F7.2,1X)
do i=1, nWindows
DO j=1,NPHAS
phase and stokes flags for I, V, Q, U
also ignored
WRITE(15,154) PHASE,(isflag(j,i,K),K=1,4)
FORMAT(F8.3,4I4,1x)
ENDDO
enddo
155
WRITE(15,155) RAID, 0.0, 2 ,0.0, vsini/1E5, vmic/1E5, KPL
FORMAT(F8.1,F10.2,I4,F10.2,2X,F6.1,’E+05’,F6.1,’E+05’,I4)
161
write the magnetic field parameters.
write(15,161) BETAD,BPDD,ADD,BQ,BOCT,razang
FORMAT(F8.1, 2X, SPF9.0, 1X, SF7.1, 3x, SPF10.1, F10.1, SF9.2)
c
213
APPENDIX B. COMPUTER PROGRAMS WRITTEN
c
got lazy here. No Idea what any of these are exept for nEle
write(15,171) nEle
format(i2," 0.001
00 1 1
")
171
c
The Elements
do i=1,nEle
1
181
write(15,181) int(abun(i,1)),(abun(i,J),J=2,9),tau0l(i),
tau0u(i),delta_abun(i)
FORMAT(I2,7F8.3,F8.2,2F8.3,SPF8.1)
enddo
WRITE(15,101) MU,frh,frhe,comment
C
112
The Model Atmosphere
do i=1,MU
WRITE(15,112) AMASS(i),T(i),RNE(i),RNATK(i),RHO(i)
FORMAT(ES16.9,’, ’,F8.1,’,’,3(ES12.5,’,’))
enddo
close(15)
111
format(1X)
END
214